Properties

Label 10-29e5-1.1-c3e5-0-0
Degree $10$
Conductor $20511149$
Sign $1$
Analytic cond. $14.6662$
Root an. cond. $1.30807$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·3-s − 7·4-s + 10·5-s + 40·7-s − 28·8-s − 19·9-s + 12·11-s − 56·12-s + 14·13-s + 80·15-s + 9·16-s + 66·17-s + 214·19-s − 70·20-s + 320·21-s + 164·23-s − 224·24-s − 159·25-s − 274·27-s − 280·28-s − 145·29-s + 420·31-s + 200·32-s + 96·33-s + 400·35-s + 133·36-s + 378·37-s + ⋯
L(s)  = 1  + 1.53·3-s − 7/8·4-s + 0.894·5-s + 2.15·7-s − 1.23·8-s − 0.703·9-s + 0.328·11-s − 1.34·12-s + 0.298·13-s + 1.37·15-s + 9/64·16-s + 0.941·17-s + 2.58·19-s − 0.782·20-s + 3.32·21-s + 1.48·23-s − 1.90·24-s − 1.27·25-s − 1.95·27-s − 1.88·28-s − 0.928·29-s + 2.43·31-s + 1.10·32-s + 0.506·33-s + 1.93·35-s + 0.615·36-s + 1.67·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20511149 ^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20511149 ^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(20511149\)    =    \(29^{5}\)
Sign: $1$
Analytic conductor: \(14.6662\)
Root analytic conductor: \(1.30807\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 20511149,\ (\ :3/2, 3/2, 3/2, 3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.105436805\)
\(L(\frac12)\) \(\approx\) \(3.105436805\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad29$C_1$ \( ( 1 + p T )^{5} \)
good2$C_2 \wr S_5$ \( 1 + 7 T^{2} + 7 p^{2} T^{3} + 5 p^{3} T^{4} + 3 p^{6} T^{5} + 5 p^{6} T^{6} + 7 p^{8} T^{7} + 7 p^{9} T^{8} + p^{15} T^{10} \)
3$C_2 \wr S_5$ \( 1 - 8 T + 83 T^{2} - 542 T^{3} + 3265 T^{4} - 18646 T^{5} + 3265 p^{3} T^{6} - 542 p^{6} T^{7} + 83 p^{9} T^{8} - 8 p^{12} T^{9} + p^{15} T^{10} \)
5$C_2 \wr S_5$ \( 1 - 2 p T + 259 T^{2} - 2096 T^{3} + 40453 T^{4} - 267034 T^{5} + 40453 p^{3} T^{6} - 2096 p^{6} T^{7} + 259 p^{9} T^{8} - 2 p^{13} T^{9} + p^{15} T^{10} \)
7$C_2 \wr S_5$ \( 1 - 40 T + 257 p T^{2} - 44112 T^{3} + 1162638 T^{4} - 20604944 T^{5} + 1162638 p^{3} T^{6} - 44112 p^{6} T^{7} + 257 p^{10} T^{8} - 40 p^{12} T^{9} + p^{15} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 12 T + 1763 T^{2} - 13714 T^{3} + 2580641 T^{4} + 37008754 T^{5} + 2580641 p^{3} T^{6} - 13714 p^{6} T^{7} + 1763 p^{9} T^{8} - 12 p^{12} T^{9} + p^{15} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 14 T + 3427 T^{2} + 10280 T^{3} - 252123 T^{4} + 167242554 T^{5} - 252123 p^{3} T^{6} + 10280 p^{6} T^{7} + 3427 p^{9} T^{8} - 14 p^{12} T^{9} + p^{15} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 66 T + 22121 T^{2} - 1091584 T^{3} + 201659462 T^{4} - 7519809404 T^{5} + 201659462 p^{3} T^{6} - 1091584 p^{6} T^{7} + 22121 p^{9} T^{8} - 66 p^{12} T^{9} + p^{15} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 214 T + 49431 T^{2} - 6213576 T^{3} + 780820954 T^{4} - 65082757348 T^{5} + 780820954 p^{3} T^{6} - 6213576 p^{6} T^{7} + 49431 p^{9} T^{8} - 214 p^{12} T^{9} + p^{15} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 164 T + 42023 T^{2} - 4665552 T^{3} + 817883710 T^{4} - 72913879960 T^{5} + 817883710 p^{3} T^{6} - 4665552 p^{6} T^{7} + 42023 p^{9} T^{8} - 164 p^{12} T^{9} + p^{15} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 420 T + 194507 T^{2} - 51412234 T^{3} + 12940581345 T^{4} - 2317742728494 T^{5} + 12940581345 p^{3} T^{6} - 51412234 p^{6} T^{7} + 194507 p^{9} T^{8} - 420 p^{12} T^{9} + p^{15} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 378 T + 183473 T^{2} - 45668424 T^{3} + 12598747514 T^{4} - 2663232080572 T^{5} + 12598747514 p^{3} T^{6} - 45668424 p^{6} T^{7} + 183473 p^{9} T^{8} - 378 p^{12} T^{9} + p^{15} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 1158 T + 807513 T^{2} + 393665952 T^{3} + 147623041214 T^{4} + 43322074461428 T^{5} + 147623041214 p^{3} T^{6} + 393665952 p^{6} T^{7} + 807513 p^{9} T^{8} + 1158 p^{12} T^{9} + p^{15} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 204 T + 303147 T^{2} + 53162326 T^{3} + 42555587009 T^{4} + 6073081393458 T^{5} + 42555587009 p^{3} T^{6} + 53162326 p^{6} T^{7} + 303147 p^{9} T^{8} + 204 p^{12} T^{9} + p^{15} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 248 T + 186419 T^{2} - 26483122 T^{3} + 12346963729 T^{4} - 356058792474 T^{5} + 12346963729 p^{3} T^{6} - 26483122 p^{6} T^{7} + 186419 p^{9} T^{8} - 248 p^{12} T^{9} + p^{15} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 554 T + 706051 T^{2} + 257171096 T^{3} + 190377537541 T^{4} + 51228756286434 T^{5} + 190377537541 p^{3} T^{6} + 257171096 p^{6} T^{7} + 706051 p^{9} T^{8} + 554 p^{12} T^{9} + p^{15} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 440 T + 931027 T^{2} - 318478640 T^{3} + 358630287294 T^{4} - 93589746901040 T^{5} + 358630287294 p^{3} T^{6} - 318478640 p^{6} T^{7} + 931027 p^{9} T^{8} - 440 p^{12} T^{9} + p^{15} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 618 T + 930749 T^{2} - 354654304 T^{3} + 335118575446 T^{4} - 95179696680076 T^{5} + 335118575446 p^{3} T^{6} - 354654304 p^{6} T^{7} + 930749 p^{9} T^{8} - 618 p^{12} T^{9} + p^{15} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 1164 T + 1251831 T^{2} - 943021136 T^{3} + 702489951274 T^{4} - 395972688310152 T^{5} + 702489951274 p^{3} T^{6} - 943021136 p^{6} T^{7} + 1251831 p^{9} T^{8} - 1164 p^{12} T^{9} + p^{15} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 692 T + 912707 T^{2} + 418495168 T^{3} + 492751147498 T^{4} + 220618574155288 T^{5} + 492751147498 p^{3} T^{6} + 418495168 p^{6} T^{7} + 912707 p^{9} T^{8} + 692 p^{12} T^{9} + p^{15} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 1950 T + 3113117 T^{2} + 3302203032 T^{3} + 2895801534794 T^{4} + 1979029953150004 T^{5} + 2895801534794 p^{3} T^{6} + 3302203032 p^{6} T^{7} + 3113117 p^{9} T^{8} + 1950 p^{12} T^{9} + p^{15} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 272 T + 967307 T^{2} + 8685042 T^{3} + 758497584129 T^{4} - 100158281651838 T^{5} + 758497584129 p^{3} T^{6} + 8685042 p^{6} T^{7} + 967307 p^{9} T^{8} - 272 p^{12} T^{9} + p^{15} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 512 T + 2338779 T^{2} - 1052487024 T^{3} + 2462969890974 T^{4} - 862752231073696 T^{5} + 2462969890974 p^{3} T^{6} - 1052487024 p^{6} T^{7} + 2338779 p^{9} T^{8} - 512 p^{12} T^{9} + p^{15} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 866 T + 2672425 T^{2} - 2372819264 T^{3} + 3305080150846 T^{4} - 2463206457135612 T^{5} + 3305080150846 p^{3} T^{6} - 2372819264 p^{6} T^{7} + 2672425 p^{9} T^{8} - 866 p^{12} T^{9} + p^{15} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 1562 T + 4976353 T^{2} - 5411957216 T^{3} + 9379455939582 T^{4} - 7296970015174028 T^{5} + 9379455939582 p^{3} T^{6} - 5411957216 p^{6} T^{7} + 4976353 p^{9} T^{8} - 1562 p^{12} T^{9} + p^{15} T^{10} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.99340376908580261400617407287, −10.16055409360588271217920302663, −9.882911262943688728693393399245, −9.778288583519302642994913067735, −9.669746484760775992390665948557, −9.228237495972259740008130201114, −8.816096839219753719756330312768, −8.609677815201642293418314790852, −8.382639295484167416013900232020, −8.381493237617127174222673623118, −7.83466147887615668445358674634, −7.69504444644151363594927333550, −7.29484990189644824066471281981, −6.50806830328555464756347889661, −6.38604761076389158484777629029, −5.88711415511282630546472448601, −5.23469622343663157608383538067, −5.20026529321895261998755933087, −5.07768903673597639646621612413, −4.37230255932457857745662798495, −3.38297875184629213312704840227, −3.30062300975049291619295067950, −2.94851289133192936358610149693, −2.07557538625614837153999979269, −1.23122749921197723148865884312, 1.23122749921197723148865884312, 2.07557538625614837153999979269, 2.94851289133192936358610149693, 3.30062300975049291619295067950, 3.38297875184629213312704840227, 4.37230255932457857745662798495, 5.07768903673597639646621612413, 5.20026529321895261998755933087, 5.23469622343663157608383538067, 5.88711415511282630546472448601, 6.38604761076389158484777629029, 6.50806830328555464756347889661, 7.29484990189644824066471281981, 7.69504444644151363594927333550, 7.83466147887615668445358674634, 8.381493237617127174222673623118, 8.382639295484167416013900232020, 8.609677815201642293418314790852, 8.816096839219753719756330312768, 9.228237495972259740008130201114, 9.669746484760775992390665948557, 9.778288583519302642994913067735, 9.882911262943688728693393399245, 10.16055409360588271217920302663, 10.99340376908580261400617407287

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