Properties

Label 4-55e4-1.1-c1e2-0-0
Degree $4$
Conductor $9150625$
Sign $1$
Analytic cond. $583.451$
Root an. cond. $4.91474$
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  + 2·2-s + 4-s − 4·7-s + 2·9-s − 8·13-s − 8·14-s + 16-s + 8·17-s + 4·18-s − 16·26-s − 4·28-s − 4·29-s − 2·32-s + 16·34-s + 2·36-s + 4·37-s − 12·41-s − 12·43-s − 2·49-s − 8·52-s − 12·53-s − 8·58-s − 8·59-s − 4·61-s − 8·63-s − 11·64-s − 8·67-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s − 1.51·7-s + 2/3·9-s − 2.21·13-s − 2.13·14-s + 1/4·16-s + 1.94·17-s + 0.942·18-s − 3.13·26-s − 0.755·28-s − 0.742·29-s − 0.353·32-s + 2.74·34-s + 1/3·36-s + 0.657·37-s − 1.87·41-s − 1.82·43-s − 2/7·49-s − 1.10·52-s − 1.64·53-s − 1.05·58-s − 1.04·59-s − 0.512·61-s − 1.00·63-s − 1.37·64-s − 0.977·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9150625\)    =    \(5^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(583.451\)
Root analytic conductor: \(4.91474\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9150625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5642406918\)
\(L(\frac12)\) \(\approx\) \(0.5642406918\)
\(L(\frac{3}{2})\) 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)$
bad5 \( 1 \)
11 \( 1 \)
good2$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 8 T + 154 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−8.959324892543342319722635787011, −8.543595794065615730625616272981, −7.86162874918331236731681670897, −7.74381476568528617341772707579, −7.23341145557558156589649864742, −7.09444665063039379876237783115, −6.42934232509989407182661143997, −6.34881421181161546653352670322, −5.56779488258114661148935208341, −5.48530395293167692716477900078, −4.98105111503882766512188745663, −4.65277066717994482718023454751, −4.33676066732336828252637210804, −3.77809346775746328276417327705, −3.23783124460662579368168011986, −3.10348990680387451660398411655, −2.78585550565994987651273464998, −1.67354856464463657761904881375, −1.58298745393235635248050307252, −0.17599655114394828323577810997, 0.17599655114394828323577810997, 1.58298745393235635248050307252, 1.67354856464463657761904881375, 2.78585550565994987651273464998, 3.10348990680387451660398411655, 3.23783124460662579368168011986, 3.77809346775746328276417327705, 4.33676066732336828252637210804, 4.65277066717994482718023454751, 4.98105111503882766512188745663, 5.48530395293167692716477900078, 5.56779488258114661148935208341, 6.34881421181161546653352670322, 6.42934232509989407182661143997, 7.09444665063039379876237783115, 7.23341145557558156589649864742, 7.74381476568528617341772707579, 7.86162874918331236731681670897, 8.543595794065615730625616272981, 8.959324892543342319722635787011

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