Dirichlet series
| L(s) = 1 | + 7·2-s + 3·3-s + 28·4-s + 5·5-s + 21·6-s + 2·7-s + 84·8-s − 2·9-s + 35·10-s + 2·11-s + 84·12-s + 5·13-s + 14·14-s + 15·15-s + 210·16-s + 8·17-s − 14·18-s + 4·19-s + 140·20-s + 6·21-s + 14·22-s − 8·23-s + 252·24-s − 2·25-s + 35·26-s − 18·27-s + 56·28-s + ⋯ |
| L(s) = 1 | + 4.94·2-s + 1.73·3-s + 14·4-s + 2.23·5-s + 8.57·6-s + 0.755·7-s + 29.6·8-s − 2/3·9-s + 11.0·10-s + 0.603·11-s + 24.2·12-s + 1.38·13-s + 3.74·14-s + 3.87·15-s + 52.5·16-s + 1.94·17-s − 3.29·18-s + 0.917·19-s + 31.3·20-s + 1.30·21-s + 2.98·22-s − 1.66·23-s + 51.4·24-s − 2/5·25-s + 6.86·26-s − 3.46·27-s + 10.5·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 193^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 193^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{7} \cdot 193^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2642.71\) |
| Root analytic conductor: | \(1.75562\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{7} \cdot 193^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(330.0743791\) |
| \(L(\frac12)\) | \(\approx\) | \(330.0743791\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{7} \) |
| 193 | \( ( 1 + T )^{7} \) | |
| good | 3 | \( 1 - p T + 11 T^{2} - 7 p T^{3} + 53 T^{4} - 100 T^{5} + 8 p^{3} T^{6} - 368 T^{7} + 8 p^{4} T^{8} - 100 p^{2} T^{9} + 53 p^{3} T^{10} - 7 p^{5} T^{11} + 11 p^{5} T^{12} - p^{7} T^{13} + p^{7} T^{14} \) |
| 5 | \( 1 - p T + 27 T^{2} - 91 T^{3} + 339 T^{4} - 922 T^{5} + 2588 T^{6} - 5636 T^{7} + 2588 p T^{8} - 922 p^{2} T^{9} + 339 p^{3} T^{10} - 91 p^{4} T^{11} + 27 p^{5} T^{12} - p^{7} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 - 2 T + 25 T^{2} - 44 T^{3} + 341 T^{4} - 66 p T^{5} + 443 p T^{6} - 3656 T^{7} + 443 p^{2} T^{8} - 66 p^{3} T^{9} + 341 p^{3} T^{10} - 44 p^{4} T^{11} + 25 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - 2 T + 41 T^{2} - 116 T^{3} + 817 T^{4} - 3022 T^{5} + 10977 T^{6} - 3944 p T^{7} + 10977 p T^{8} - 3022 p^{2} T^{9} + 817 p^{3} T^{10} - 116 p^{4} T^{11} + 41 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 5 T + 35 T^{2} - 141 T^{3} + 697 T^{4} - 2416 T^{5} + 10670 T^{6} - 34052 T^{7} + 10670 p T^{8} - 2416 p^{2} T^{9} + 697 p^{3} T^{10} - 141 p^{4} T^{11} + 35 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 8 T + 83 T^{2} - 392 T^{3} + 2377 T^{4} - 440 p T^{5} + 38419 T^{6} - 106192 T^{7} + 38419 p T^{8} - 440 p^{3} T^{9} + 2377 p^{3} T^{10} - 392 p^{4} T^{11} + 83 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 4 T + 97 T^{2} - 312 T^{3} + 4409 T^{4} - 11676 T^{5} + 124097 T^{6} - 272432 T^{7} + 124097 p T^{8} - 11676 p^{2} T^{9} + 4409 p^{3} T^{10} - 312 p^{4} T^{11} + 97 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 8 T + 125 T^{2} + 704 T^{3} + 6345 T^{4} + 27704 T^{5} + 194381 T^{6} + 724864 T^{7} + 194381 p T^{8} + 27704 p^{2} T^{9} + 6345 p^{3} T^{10} + 704 p^{4} T^{11} + 125 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 127 T^{2} + 184 T^{3} + 7329 T^{4} + 20080 T^{5} + 272087 T^{6} + 854512 T^{7} + 272087 p T^{8} + 20080 p^{2} T^{9} + 7329 p^{3} T^{10} + 184 p^{4} T^{11} + 127 p^{5} T^{12} + p^{7} T^{14} \) | |
| 31 | \( 1 + 6 T + 165 T^{2} + 820 T^{3} + 12593 T^{4} + 53370 T^{5} + 591837 T^{6} + 2092440 T^{7} + 591837 p T^{8} + 53370 p^{2} T^{9} + 12593 p^{3} T^{10} + 820 p^{4} T^{11} + 165 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + T + 69 T^{2} + 3 p T^{3} + 4609 T^{4} + 5568 T^{5} + 5564 p T^{6} + 375304 T^{7} + 5564 p^{2} T^{8} + 5568 p^{2} T^{9} + 4609 p^{3} T^{10} + 3 p^{5} T^{11} + 69 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 4 T + 39 T^{2} + 352 T^{3} + 3141 T^{4} + 28364 T^{5} + 167251 T^{6} + 812544 T^{7} + 167251 p T^{8} + 28364 p^{2} T^{9} + 3141 p^{3} T^{10} + 352 p^{4} T^{11} + 39 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 13 T + 153 T^{2} + 21 p T^{3} + 4133 T^{4} - 13376 T^{5} - 238896 T^{6} - 2565516 T^{7} - 238896 p T^{8} - 13376 p^{2} T^{9} + 4133 p^{3} T^{10} + 21 p^{5} T^{11} + 153 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 17 T + 413 T^{2} + 4831 T^{3} + 65451 T^{4} + 568754 T^{5} + 5392846 T^{6} + 35649348 T^{7} + 5392846 p T^{8} + 568754 p^{2} T^{9} + 65451 p^{3} T^{10} + 4831 p^{4} T^{11} + 413 p^{5} T^{12} + 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 5 T + 173 T^{2} + 1061 T^{3} + 17999 T^{4} + 101830 T^{5} + 1261118 T^{6} + 6697192 T^{7} + 1261118 p T^{8} + 101830 p^{2} T^{9} + 17999 p^{3} T^{10} + 1061 p^{4} T^{11} + 173 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 3 T + 165 T^{2} + 709 T^{3} + 15557 T^{4} + 69444 T^{5} + 1242072 T^{6} + 4526408 T^{7} + 1242072 p T^{8} + 69444 p^{2} T^{9} + 15557 p^{3} T^{10} + 709 p^{4} T^{11} + 165 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 12 T + 251 T^{2} - 2592 T^{3} + 34029 T^{4} - 313460 T^{5} + 2993759 T^{6} - 23322432 T^{7} + 2993759 p T^{8} - 313460 p^{2} T^{9} + 34029 p^{3} T^{10} - 2592 p^{4} T^{11} + 251 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - T + 247 T^{2} - 623 T^{3} + 32749 T^{4} - 98876 T^{5} + 2920656 T^{6} - 8920528 T^{7} + 2920656 p T^{8} - 98876 p^{2} T^{9} + 32749 p^{3} T^{10} - 623 p^{4} T^{11} + 247 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 25 T + 615 T^{2} + 9517 T^{3} + 138713 T^{4} + 1570652 T^{5} + 16674380 T^{6} + 145182940 T^{7} + 16674380 p T^{8} + 1570652 p^{2} T^{9} + 138713 p^{3} T^{10} + 9517 p^{4} T^{11} + 615 p^{5} T^{12} + 25 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 28 T + 587 T^{2} - 8320 T^{3} + 106425 T^{4} - 1138692 T^{5} + 11663995 T^{6} - 102631264 T^{7} + 11663995 p T^{8} - 1138692 p^{2} T^{9} + 106425 p^{3} T^{10} - 8320 p^{4} T^{11} + 587 p^{5} T^{12} - 28 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 5 T + 353 T^{2} + 1579 T^{3} + 63597 T^{4} + 243244 T^{5} + 7331166 T^{6} + 23823596 T^{7} + 7331166 p T^{8} + 243244 p^{2} T^{9} + 63597 p^{3} T^{10} + 1579 p^{4} T^{11} + 353 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 7 T + 353 T^{2} + 2885 T^{3} + 63529 T^{4} + 523936 T^{5} + 7525244 T^{6} + 55323360 T^{7} + 7525244 p T^{8} + 523936 p^{2} T^{9} + 63529 p^{3} T^{10} + 2885 p^{4} T^{11} + 353 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 4 T + 467 T^{2} - 1976 T^{3} + 102721 T^{4} - 423676 T^{5} + 13802795 T^{6} - 49640688 T^{7} + 13802795 p T^{8} - 423676 p^{2} T^{9} + 102721 p^{3} T^{10} - 1976 p^{4} T^{11} + 467 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 13 T + 389 T^{2} - 4037 T^{3} + 65569 T^{4} - 515934 T^{5} + 7013182 T^{6} - 47954198 T^{7} + 7013182 p T^{8} - 515934 p^{2} T^{9} + 65569 p^{3} T^{10} - 4037 p^{4} T^{11} + 389 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.57014182200238104876202183917, −5.27435877656201555663411601788, −5.20910398126148343288133236321, −5.20827368796659717081060456866, −4.90761193496094065963557700262, −4.85068879749366785367785034748, −4.42713221359685122196530750027, −4.30635229607659871713254217155, −4.22889557206238942359971549891, −3.99426117631620890879580659796, −3.86604224831251629102821668237, −3.54773461965579110452358403626, −3.37118257688913539100810834542, −3.24283951984012850859194745343, −3.20284994198936883913667240359, −3.18613735272996313193709947488, −3.00267279038553993853517370573, −2.87476209244852239539635589514, −2.12760258744898540407365226767, −2.07031133644891265952137298990, −2.04917841485763769803704814069, −1.89881035714889866011265481715, −1.89423310519365375569739906382, −1.40857122438615077263995550570, −1.27934994704796745932215206811, 1.27934994704796745932215206811, 1.40857122438615077263995550570, 1.89423310519365375569739906382, 1.89881035714889866011265481715, 2.04917841485763769803704814069, 2.07031133644891265952137298990, 2.12760258744898540407365226767, 2.87476209244852239539635589514, 3.00267279038553993853517370573, 3.18613735272996313193709947488, 3.20284994198936883913667240359, 3.24283951984012850859194745343, 3.37118257688913539100810834542, 3.54773461965579110452358403626, 3.86604224831251629102821668237, 3.99426117631620890879580659796, 4.22889557206238942359971549891, 4.30635229607659871713254217155, 4.42713221359685122196530750027, 4.85068879749366785367785034748, 4.90761193496094065963557700262, 5.20827368796659717081060456866, 5.20910398126148343288133236321, 5.27435877656201555663411601788, 5.57014182200238104876202183917
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.