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Properties

Label 14-4970e7-1.1-c1e7-0-3
Degree $14$
Conductor $7.490\times 10^{25}$
Sign $-1$
Analytic cond. $1.55036\times 10^{11}$
Root an. cond. $6.29965$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $7$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 7·2-s + 3-s + 28·4-s − 7·5-s + 7·6-s − 7·7-s + 84·8-s − 9·9-s − 49·10-s − 9·11-s + 28·12-s − 49·14-s − 7·15-s + 210·16-s − 5·17-s − 63·18-s + 2·19-s − 196·20-s − 7·21-s − 63·22-s − 8·23-s + 84·24-s + 28·25-s − 10·27-s − 196·28-s − 19·29-s − 49·30-s + ⋯
L(s)  = 1  + 4.94·2-s + 0.577·3-s + 14·4-s − 3.13·5-s + 2.85·6-s − 2.64·7-s + 29.6·8-s − 3·9-s − 15.4·10-s − 2.71·11-s + 8.08·12-s − 13.0·14-s − 1.80·15-s + 52.5·16-s − 1.21·17-s − 14.8·18-s + 0.458·19-s − 43.8·20-s − 1.52·21-s − 13.4·22-s − 1.66·23-s + 17.1·24-s + 28/5·25-s − 1.92·27-s − 37.0·28-s − 3.52·29-s − 8.94·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 5^{7} \cdot 7^{7} \cdot 71^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 5^{7} \cdot 7^{7} \cdot 71^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(14\)
Conductor: \(2^{7} \cdot 5^{7} \cdot 7^{7} \cdot 71^{7}\)
Sign: $-1$
Analytic conductor: \(1.55036\times 10^{11}\)
Root analytic conductor: \(6.29965\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(7\)
Selberg data: \((14,\ 2^{7} \cdot 5^{7} \cdot 7^{7} \cdot 71^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T )^{7} \)
5 \( ( 1 + T )^{7} \)
7 \( ( 1 + T )^{7} \)
71 \( ( 1 - T )^{7} \)
good3 \( 1 - T + 10 T^{2} - p^{2} T^{3} + 59 T^{4} - 17 p T^{5} + 79 p T^{6} - 176 T^{7} + 79 p^{2} T^{8} - 17 p^{3} T^{9} + 59 p^{3} T^{10} - p^{6} T^{11} + 10 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \)
11 \( 1 + 9 T + 93 T^{2} + 554 T^{3} + 3293 T^{4} + 14483 T^{5} + 61700 T^{6} + 208515 T^{7} + 61700 p T^{8} + 14483 p^{2} T^{9} + 3293 p^{3} T^{10} + 554 p^{4} T^{11} + 93 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \)
13 \( 1 + 47 T^{2} + 4 T^{3} + 1005 T^{4} + 233 T^{5} + 14372 T^{6} + 4575 T^{7} + 14372 p T^{8} + 233 p^{2} T^{9} + 1005 p^{3} T^{10} + 4 p^{4} T^{11} + 47 p^{5} T^{12} + p^{7} T^{14} \)
17 \( 1 + 5 T + 40 T^{2} + 147 T^{3} + 1269 T^{4} + 4483 T^{5} + 26846 T^{6} + 76522 T^{7} + 26846 p T^{8} + 4483 p^{2} T^{9} + 1269 p^{3} T^{10} + 147 p^{4} T^{11} + 40 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \)
19 \( 1 - 2 T + 71 T^{2} + 18 T^{3} + 1551 T^{4} + 7689 T^{5} + 8353 T^{6} + 251794 T^{7} + 8353 p T^{8} + 7689 p^{2} T^{9} + 1551 p^{3} T^{10} + 18 p^{4} T^{11} + 71 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \)
23 \( 1 + 8 T + 129 T^{2} + 692 T^{3} + 6679 T^{4} + 26991 T^{5} + 209028 T^{6} + 706575 T^{7} + 209028 p T^{8} + 26991 p^{2} T^{9} + 6679 p^{3} T^{10} + 692 p^{4} T^{11} + 129 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \)
29 \( 1 + 19 T + 9 p T^{2} + 2612 T^{3} + 23061 T^{4} + 169662 T^{5} + 1112474 T^{6} + 6287758 T^{7} + 1112474 p T^{8} + 169662 p^{2} T^{9} + 23061 p^{3} T^{10} + 2612 p^{4} T^{11} + 9 p^{6} T^{12} + 19 p^{6} T^{13} + p^{7} T^{14} \)
31 \( 1 + 6 T + 167 T^{2} + 866 T^{3} + 12717 T^{4} + 58017 T^{5} + 592359 T^{6} + 2291626 T^{7} + 592359 p T^{8} + 58017 p^{2} T^{9} + 12717 p^{3} T^{10} + 866 p^{4} T^{11} + 167 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \)
37 \( 1 + 10 T + 216 T^{2} + 1646 T^{3} + 21132 T^{4} + 130201 T^{5} + 1224295 T^{6} + 6099667 T^{7} + 1224295 p T^{8} + 130201 p^{2} T^{9} + 21132 p^{3} T^{10} + 1646 p^{4} T^{11} + 216 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \)
41 \( 1 + 19 T + 338 T^{2} + 4041 T^{3} + 43126 T^{4} + 376589 T^{5} + 2950991 T^{6} + 19912606 T^{7} + 2950991 p T^{8} + 376589 p^{2} T^{9} + 43126 p^{3} T^{10} + 4041 p^{4} T^{11} + 338 p^{5} T^{12} + 19 p^{6} T^{13} + p^{7} T^{14} \)
43 \( 1 + 22 T + 446 T^{2} + 5708 T^{3} + 67247 T^{4} + 609531 T^{5} + 5120266 T^{6} + 34846694 T^{7} + 5120266 p T^{8} + 609531 p^{2} T^{9} + 67247 p^{3} T^{10} + 5708 p^{4} T^{11} + 446 p^{5} T^{12} + 22 p^{6} T^{13} + p^{7} T^{14} \)
47 \( 1 - 2 T + 154 T^{2} - 598 T^{3} + 11177 T^{4} - 75540 T^{5} + 572944 T^{6} - 4857536 T^{7} + 572944 p T^{8} - 75540 p^{2} T^{9} + 11177 p^{3} T^{10} - 598 p^{4} T^{11} + 154 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \)
53 \( 1 + 3 T + 152 T^{2} - 19 T^{3} + 11899 T^{4} - 31173 T^{5} + 712408 T^{6} - 2399150 T^{7} + 712408 p T^{8} - 31173 p^{2} T^{9} + 11899 p^{3} T^{10} - 19 p^{4} T^{11} + 152 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \)
59 \( 1 - 13 T + 408 T^{2} - 4157 T^{3} + 71259 T^{4} - 576431 T^{5} + 6950496 T^{6} - 44413486 T^{7} + 6950496 p T^{8} - 576431 p^{2} T^{9} + 71259 p^{3} T^{10} - 4157 p^{4} T^{11} + 408 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \)
61 \( 1 + 4 T + 186 T^{2} + 1021 T^{3} + 22263 T^{4} + 111393 T^{5} + 1882483 T^{6} + 8398484 T^{7} + 1882483 p T^{8} + 111393 p^{2} T^{9} + 22263 p^{3} T^{10} + 1021 p^{4} T^{11} + 186 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \)
67 \( 1 + 23 T + 516 T^{2} + 7329 T^{3} + 101669 T^{4} + 1082415 T^{5} + 11119334 T^{6} + 92509634 T^{7} + 11119334 p T^{8} + 1082415 p^{2} T^{9} + 101669 p^{3} T^{10} + 7329 p^{4} T^{11} + 516 p^{5} T^{12} + 23 p^{6} T^{13} + p^{7} T^{14} \)
73 \( 1 + 228 T^{2} - 666 T^{3} + 28661 T^{4} - 105405 T^{5} + 2960826 T^{6} - 8309494 T^{7} + 2960826 p T^{8} - 105405 p^{2} T^{9} + 28661 p^{3} T^{10} - 666 p^{4} T^{11} + 228 p^{5} T^{12} + p^{7} T^{14} \)
79 \( 1 + 40 T + 952 T^{2} + 15530 T^{3} + 202183 T^{4} + 2202243 T^{5} + 21756964 T^{6} + 198076962 T^{7} + 21756964 p T^{8} + 2202243 p^{2} T^{9} + 202183 p^{3} T^{10} + 15530 p^{4} T^{11} + 952 p^{5} T^{12} + 40 p^{6} T^{13} + p^{7} T^{14} \)
83 \( 1 - 24 T + 511 T^{2} - 8110 T^{3} + 112931 T^{4} - 1336144 T^{5} + 14396100 T^{6} - 137444998 T^{7} + 14396100 p T^{8} - 1336144 p^{2} T^{9} + 112931 p^{3} T^{10} - 8110 p^{4} T^{11} + 511 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \)
89 \( 1 + 4 T + 301 T^{2} + 1020 T^{3} + 41592 T^{4} + 165108 T^{5} + 4137418 T^{6} + 18660312 T^{7} + 4137418 p T^{8} + 165108 p^{2} T^{9} + 41592 p^{3} T^{10} + 1020 p^{4} T^{11} + 301 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \)
97 \( 1 + 35 T + 1075 T^{2} + 21194 T^{3} + 374925 T^{4} + 5152622 T^{5} + 64668664 T^{6} + 664485290 T^{7} + 64668664 p T^{8} + 5152622 p^{2} T^{9} + 374925 p^{3} T^{10} + 21194 p^{4} T^{11} + 1075 p^{5} T^{12} + 35 p^{6} T^{13} + p^{7} T^{14} \)
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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

−4.07263682670243197600297259536, −3.90633288887653128288810843016, −3.90166558067085938908738658736, −3.83517741834683903227444233289, −3.66224235314226211813074331739, −3.52427756799381031979255120065, −3.52048912347807015699151012114, −3.41872681266262532071795943832, −3.30410795871667282010206013687, −3.21699834777175609600847363960, −3.12409186667214147218670368294, −2.78454918863538535175458794901, −2.71762873051039933190187524981, −2.67621927585329197284133239335, −2.63039209048203690438045535468, −2.59852104808733597302744144312, −2.49854272344665814847869680564, −2.44291069960212244769176977396, −2.01573020617187281406291729892, −1.92213135208658008168804274688, −1.68699646045825452648934940972, −1.62880688308421555622070964649, −1.26762665413203278989207382941, −1.25275724004094831450828135653, −1.19964713376293945521917128654, 0, 0, 0, 0, 0, 0, 0, 1.19964713376293945521917128654, 1.25275724004094831450828135653, 1.26762665413203278989207382941, 1.62880688308421555622070964649, 1.68699646045825452648934940972, 1.92213135208658008168804274688, 2.01573020617187281406291729892, 2.44291069960212244769176977396, 2.49854272344665814847869680564, 2.59852104808733597302744144312, 2.63039209048203690438045535468, 2.67621927585329197284133239335, 2.71762873051039933190187524981, 2.78454918863538535175458794901, 3.12409186667214147218670368294, 3.21699834777175609600847363960, 3.30410795871667282010206013687, 3.41872681266262532071795943832, 3.52048912347807015699151012114, 3.52427756799381031979255120065, 3.66224235314226211813074331739, 3.83517741834683903227444233289, 3.90166558067085938908738658736, 3.90633288887653128288810843016, 4.07263682670243197600297259536

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.