Properties

Label 4-8470e2-1.1-c1e2-0-12
Degree $4$
Conductor $71740900$
Sign $1$
Analytic cond. $4574.26$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 3·4-s + 2·5-s + 4·6-s + 2·7-s − 4·8-s + 2·9-s − 4·10-s − 6·12-s − 4·14-s − 4·15-s + 5·16-s − 4·17-s − 4·18-s + 6·19-s + 6·20-s − 4·21-s − 10·23-s + 8·24-s + 3·25-s − 6·27-s + 6·28-s − 2·29-s + 8·30-s − 6·32-s + 8·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s + 0.755·7-s − 1.41·8-s + 2/3·9-s − 1.26·10-s − 1.73·12-s − 1.06·14-s − 1.03·15-s + 5/4·16-s − 0.970·17-s − 0.942·18-s + 1.37·19-s + 1.34·20-s − 0.872·21-s − 2.08·23-s + 1.63·24-s + 3/5·25-s − 1.15·27-s + 1.13·28-s − 0.371·29-s + 1.46·30-s − 1.06·32-s + 1.37·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(71740900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(4574.26\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 71740900,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 6 T + 42 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 2 T + 38 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 20 T + 202 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 6 T + 162 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
89$C_4$ \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 10 T + 174 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.65345397981329443272174860512, −7.50116432651474421314461695546, −6.92798389366552246157480133359, −6.67005623139072794795530368835, −6.03359152264425140046870394069, −5.97379576247946229508789422089, −5.70818132462652036012992251068, −5.59514040065078744570463601238, −4.72941183765268796507665457879, −4.56364495027765152557081532474, −4.31337981512904027470894697084, −3.61941718864520282469452912212, −3.00372788027609021912620474765, −2.81737158472511516893358934178, −2.00479265173189871046101222147, −1.83859757624555839716819092364, −1.47759404608975644265129384789, −0.981415258602670208691664704536, 0, 0, 0.981415258602670208691664704536, 1.47759404608975644265129384789, 1.83859757624555839716819092364, 2.00479265173189871046101222147, 2.81737158472511516893358934178, 3.00372788027609021912620474765, 3.61941718864520282469452912212, 4.31337981512904027470894697084, 4.56364495027765152557081532474, 4.72941183765268796507665457879, 5.59514040065078744570463601238, 5.70818132462652036012992251068, 5.97379576247946229508789422089, 6.03359152264425140046870394069, 6.67005623139072794795530368835, 6.92798389366552246157480133359, 7.50116432651474421314461695546, 7.65345397981329443272174860512

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