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Properties

Label 10-8036e5-1.1-c1e5-0-1
Degree $10$
Conductor $3.351\times 10^{19}$
Sign $-1$
Analytic cond. $1.08789\times 10^{9}$
Root an. cond. $8.01047$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $5$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 3·5-s − 3·9-s − 7·13-s + 6·15-s + 3·17-s − 10·19-s + 12·23-s − 7·25-s + 6·27-s − 3·29-s − 7·31-s + 37-s + 14·39-s − 5·41-s + 13·43-s + 9·45-s − 9·47-s − 6·51-s − 3·53-s + 20·57-s + 3·59-s − 16·61-s + 21·65-s + 19·67-s − 24·69-s − 12·71-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.34·5-s − 9-s − 1.94·13-s + 1.54·15-s + 0.727·17-s − 2.29·19-s + 2.50·23-s − 7/5·25-s + 1.15·27-s − 0.557·29-s − 1.25·31-s + 0.164·37-s + 2.24·39-s − 0.780·41-s + 1.98·43-s + 1.34·45-s − 1.31·47-s − 0.840·51-s − 0.412·53-s + 2.64·57-s + 0.390·59-s − 2.04·61-s + 2.60·65-s + 2.32·67-s − 2.88·69-s − 1.42·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 7^{10} \cdot 41^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 7^{10} \cdot 41^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{10} \cdot 7^{10} \cdot 41^{5}\)
Sign: $-1$
Analytic conductor: \(1.08789\times 10^{9}\)
Root analytic conductor: \(8.01047\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{10} \cdot 7^{10} \cdot 41^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41$C_1$ \( ( 1 + T )^{5} \)
good3$C_2 \wr S_5$ \( 1 + 2 T + 7 T^{2} + 14 T^{3} + 31 T^{4} + 59 T^{5} + 31 p T^{6} + 14 p^{2} T^{7} + 7 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + 3 T + 16 T^{2} + 8 p T^{3} + 127 T^{4} + 274 T^{5} + 127 p T^{6} + 8 p^{3} T^{7} + 16 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 34 T^{2} + 4 T^{3} + 601 T^{4} + 16 T^{5} + 601 p T^{6} + 4 p^{2} T^{7} + 34 p^{3} T^{8} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 7 T + 72 T^{2} + 326 T^{3} + 1883 T^{4} + 6081 T^{5} + 1883 p T^{6} + 326 p^{2} T^{7} + 72 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 3 T + 40 T^{2} - 132 T^{3} + 1135 T^{4} - 2835 T^{5} + 1135 p T^{6} - 132 p^{2} T^{7} + 40 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 10 T + 117 T^{2} + 730 T^{3} + 4783 T^{4} + 20523 T^{5} + 4783 p T^{6} + 730 p^{2} T^{7} + 117 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 12 T + 163 T^{2} - 1174 T^{3} + 8623 T^{4} - 41305 T^{5} + 8623 p T^{6} - 1174 p^{2} T^{7} + 163 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 3 T + 124 T^{2} + 284 T^{3} + 6667 T^{4} + 11690 T^{5} + 6667 p T^{6} + 284 p^{2} T^{7} + 124 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 7 T + 162 T^{2} + 842 T^{3} + 10229 T^{4} + 38742 T^{5} + 10229 p T^{6} + 842 p^{2} T^{7} + 162 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - T + 129 T^{2} - 293 T^{3} + 7484 T^{4} - 18756 T^{5} + 7484 p T^{6} - 293 p^{2} T^{7} + 129 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 13 T + 234 T^{2} - 2046 T^{3} + 20829 T^{4} - 128475 T^{5} + 20829 p T^{6} - 2046 p^{2} T^{7} + 234 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 9 T + 169 T^{2} + 1585 T^{3} + 12862 T^{4} + 109216 T^{5} + 12862 p T^{6} + 1585 p^{2} T^{7} + 169 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 3 T + 256 T^{2} + 616 T^{3} + 26671 T^{4} + 48466 T^{5} + 26671 p T^{6} + 616 p^{2} T^{7} + 256 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 3 T + 208 T^{2} - 530 T^{3} + 20587 T^{4} - 40862 T^{5} + 20587 p T^{6} - 530 p^{2} T^{7} + 208 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 16 T + 174 T^{2} + 270 T^{3} - 7827 T^{4} - 122124 T^{5} - 7827 p T^{6} + 270 p^{2} T^{7} + 174 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 19 T + 360 T^{2} - 4400 T^{3} + 48887 T^{4} - 419922 T^{5} + 48887 p T^{6} - 4400 p^{2} T^{7} + 360 p^{3} T^{8} - 19 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 12 T + 286 T^{2} + 2964 T^{3} + 35425 T^{4} + 300480 T^{5} + 35425 p T^{6} + 2964 p^{2} T^{7} + 286 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 4 T + 282 T^{2} + 914 T^{3} + 36665 T^{4} + 94644 T^{5} + 36665 p T^{6} + 914 p^{2} T^{7} + 282 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 28 T + 579 T^{2} - 8320 T^{3} + 100402 T^{4} - 956712 T^{5} + 100402 p T^{6} - 8320 p^{2} T^{7} + 579 p^{3} T^{8} - 28 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 18 T + 418 T^{2} + 4746 T^{3} + 63337 T^{4} + 530832 T^{5} + 63337 p T^{6} + 4746 p^{2} T^{7} + 418 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 217 T^{2} - 1284 T^{3} + 17623 T^{4} - 224799 T^{5} + 17623 p T^{6} - 1284 p^{2} T^{7} + 217 p^{3} T^{8} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 4 T + 405 T^{2} + 1326 T^{3} + 71967 T^{4} + 181845 T^{5} + 71967 p T^{6} + 1326 p^{2} T^{7} + 405 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.87533999076483818150130558213, −4.87354123093327141480020042413, −4.69580824123097001204072378896, −4.58181666740488020750320269215, −4.41494547825442648338172826952, −4.14843422597206700981760921689, −4.13109421683395964480608728833, −3.86874915534692554723065746883, −3.76908677275475979357526998035, −3.74003066312526913286915118054, −3.27976972315425613742145365766, −3.27964567745329965784525544889, −3.25750692028383551428871234674, −2.91600660428906794924350844145, −2.88231279182508606252620480657, −2.39330094418500818794528762166, −2.37736438033560084425350544799, −2.28399173153454719761501409978, −2.21739579887326355486513928343, −1.99254216321004473206310587596, −1.61690575657937804790536010539, −1.38023755789492698481249041305, −1.20529387162219020389485705432, −0.882653404439493735911347354947, −0.876829949932666053273863265711, 0, 0, 0, 0, 0, 0.876829949932666053273863265711, 0.882653404439493735911347354947, 1.20529387162219020389485705432, 1.38023755789492698481249041305, 1.61690575657937804790536010539, 1.99254216321004473206310587596, 2.21739579887326355486513928343, 2.28399173153454719761501409978, 2.37736438033560084425350544799, 2.39330094418500818794528762166, 2.88231279182508606252620480657, 2.91600660428906794924350844145, 3.25750692028383551428871234674, 3.27964567745329965784525544889, 3.27976972315425613742145365766, 3.74003066312526913286915118054, 3.76908677275475979357526998035, 3.86874915534692554723065746883, 4.13109421683395964480608728833, 4.14843422597206700981760921689, 4.41494547825442648338172826952, 4.58181666740488020750320269215, 4.69580824123097001204072378896, 4.87354123093327141480020042413, 4.87533999076483818150130558213

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