Properties

Label 10-3330e5-1.1-c1e5-0-0
Degree $10$
Conductor $4.095\times 10^{17}$
Sign $1$
Analytic cond. $1.32925\times 10^{7}$
Root an. cond. $5.15656$
Motivic weight $1$
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  − 5·2-s + 15·4-s − 5·5-s + 3·7-s − 35·8-s + 25·10-s − 5·11-s + 5·13-s − 15·14-s + 70·16-s − 7·17-s − 19-s − 75·20-s + 25·22-s − 5·23-s + 15·25-s − 25·26-s + 45·28-s − 2·29-s + 16·31-s − 126·32-s + 35·34-s − 15·35-s + 5·37-s + 5·38-s + 175·40-s − 2·41-s + ⋯
L(s)  = 1  − 3.53·2-s + 15/2·4-s − 2.23·5-s + 1.13·7-s − 12.3·8-s + 7.90·10-s − 1.50·11-s + 1.38·13-s − 4.00·14-s + 35/2·16-s − 1.69·17-s − 0.229·19-s − 16.7·20-s + 5.33·22-s − 1.04·23-s + 3·25-s − 4.90·26-s + 8.50·28-s − 0.371·29-s + 2.87·31-s − 22.2·32-s + 6.00·34-s − 2.53·35-s + 0.821·37-s + 0.811·38-s + 27.6·40-s − 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{10} \cdot 5^{5} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{10} \cdot 5^{5} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{5} \cdot 3^{10} \cdot 5^{5} \cdot 37^{5}\)
Sign: $1$
Analytic conductor: \(1.32925\times 10^{7}\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3330} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{5} \cdot 3^{10} \cdot 5^{5} \cdot 37^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.4445507825\)
\(L(\frac12)\) \(\approx\) \(0.4445507825\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{5} \)
3 \( 1 \)
5$C_1$ \( ( 1 + T )^{5} \)
37$C_1$ \( ( 1 - T )^{5} \)
good7$S_5\times C_2$ \( 1 - 3 T + 2 p T^{2} - 29 T^{3} + 97 T^{4} - 160 T^{5} + 97 p T^{6} - 29 p^{2} T^{7} + 2 p^{4} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 5 T + 26 T^{2} + 109 T^{3} + 463 T^{4} + 1776 T^{5} + 463 p T^{6} + 109 p^{2} T^{7} + 26 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 5 T + 33 T^{2} - 148 T^{3} + 746 T^{4} - 2526 T^{5} + 746 p T^{6} - 148 p^{2} T^{7} + 33 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 7 T + 80 T^{2} + 413 T^{3} + 155 p T^{4} + 10080 T^{5} + 155 p^{2} T^{6} + 413 p^{2} T^{7} + 80 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + T + 9 T^{2} + 116 T^{3} + 620 T^{4} + 54 T^{5} + 620 p T^{6} + 116 p^{2} T^{7} + 9 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 5 T + 83 T^{2} + 340 T^{3} + 3370 T^{4} + 10734 T^{5} + 3370 p T^{6} + 340 p^{2} T^{7} + 83 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 2 T + 110 T^{2} + 208 T^{3} + 5617 T^{4} + 8796 T^{5} + 5617 p T^{6} + 208 p^{2} T^{7} + 110 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 16 T + 192 T^{2} - 1512 T^{3} + 10431 T^{4} - 59424 T^{5} + 10431 p T^{6} - 1512 p^{2} T^{7} + 192 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 2 T + 110 T^{2} + 154 T^{3} + 6757 T^{4} + 5520 T^{5} + 6757 p T^{6} + 154 p^{2} T^{7} + 110 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 12 T + 176 T^{2} - 1448 T^{3} + 13399 T^{4} - 81640 T^{5} + 13399 p T^{6} - 1448 p^{2} T^{7} + 176 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 6 T + 115 T^{2} + 624 T^{3} + 8194 T^{4} + 44244 T^{5} + 8194 p T^{6} + 624 p^{2} T^{7} + 115 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 3 T + 214 T^{2} + 495 T^{3} + 20305 T^{4} + 35472 T^{5} + 20305 p T^{6} + 495 p^{2} T^{7} + 214 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 10 T + 179 T^{2} - 1472 T^{3} + 17638 T^{4} - 122892 T^{5} + 17638 p T^{6} - 1472 p^{2} T^{7} + 179 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 16 T + 138 T^{2} - 24 p T^{3} + 15201 T^{4} - 121128 T^{5} + 15201 p T^{6} - 24 p^{3} T^{7} + 138 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 4 T + 87 T^{2} + 528 T^{3} + 162 T^{4} + 69288 T^{5} + 162 p T^{6} + 528 p^{2} T^{7} + 87 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 8 T + 203 T^{2} - 1792 T^{3} + 23794 T^{4} - 169200 T^{5} + 23794 p T^{6} - 1792 p^{2} T^{7} + 203 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 3 T + 113 T^{2} + 688 T^{3} + 12934 T^{4} + 46682 T^{5} + 12934 p T^{6} + 688 p^{2} T^{7} + 113 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 2 T + 219 T^{2} - 88 T^{3} + 26378 T^{4} - 5676 T^{5} + 26378 p T^{6} - 88 p^{2} T^{7} + 219 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 5 T + 239 T^{2} - 2176 T^{3} + 25930 T^{4} - 290022 T^{5} + 25930 p T^{6} - 2176 p^{2} T^{7} + 239 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 7 T + 149 T^{2} - 1148 T^{3} + 17026 T^{4} - 177306 T^{5} + 17026 p T^{6} - 1148 p^{2} T^{7} + 149 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 14 T + 438 T^{2} - 5026 T^{3} + 81785 T^{4} - 710808 T^{5} + 81785 p T^{6} - 5026 p^{2} T^{7} + 438 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.96525021023895946376722601316, −4.84453982953496869275825758948, −4.77767455846619175188284764738, −4.69358193313141894089764923850, −4.65987674016933643008053859268, −3.98038723238864326603826728163, −3.94708077488087859753623687728, −3.90845768797764489811838757303, −3.84554899457031056581422792859, −3.75697525328792306958560876014, −3.11970214206761240646152488649, −2.93712978668304244517823745985, −2.83785287738202160502668448931, −2.82974217841032749273545653126, −2.75131096565439625341395241218, −2.22112357831499880577542548263, −2.02279419449629111608507366256, −1.91446810302686765118371170020, −1.77668608694996462372195969161, −1.64293343685636682041356483020, −1.02149226798114432285190621341, −0.76905522154442237512605290149, −0.75402223756590700038271023621, −0.59569632687008645347450801609, −0.25204129783875144632904080893, 0.25204129783875144632904080893, 0.59569632687008645347450801609, 0.75402223756590700038271023621, 0.76905522154442237512605290149, 1.02149226798114432285190621341, 1.64293343685636682041356483020, 1.77668608694996462372195969161, 1.91446810302686765118371170020, 2.02279419449629111608507366256, 2.22112357831499880577542548263, 2.75131096565439625341395241218, 2.82974217841032749273545653126, 2.83785287738202160502668448931, 2.93712978668304244517823745985, 3.11970214206761240646152488649, 3.75697525328792306958560876014, 3.84554899457031056581422792859, 3.90845768797764489811838757303, 3.94708077488087859753623687728, 3.98038723238864326603826728163, 4.65987674016933643008053859268, 4.69358193313141894089764923850, 4.77767455846619175188284764738, 4.84453982953496869275825758948, 4.96525021023895946376722601316

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