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Properties

Label 4-74e2-1.1-c1e2-0-2
Degree $4$
Conductor $5476$
Sign $1$
Analytic cond. $0.349154$
Root an. cond. $0.768695$
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 − 3-s + 3·4-s + 5-s − 2·6-s − 2·7-s + 4·8-s − 4·9-s + 2·10-s − 5·11-s − 3·12-s + 13-s − 4·14-s − 15-s + 5·16-s − 8·18-s + 3·20-s + 2·21-s − 10·22-s − 23-s − 4·24-s + 2·25-s + 2·26-s + 6·27-s − 6·28-s − 3·29-s − 2·30-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 3/2·4-s + 0.447·5-s − 0.816·6-s − 0.755·7-s + 1.41·8-s − 4/3·9-s + 0.632·10-s − 1.50·11-s − 0.866·12-s + 0.277·13-s − 1.06·14-s − 0.258·15-s + 5/4·16-s − 1.88·18-s + 0.670·20-s + 0.436·21-s − 2.13·22-s − 0.208·23-s − 0.816·24-s + 2/5·25-s + 0.392·26-s + 1.15·27-s − 1.13·28-s − 0.557·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5476 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5476 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5476\)    =    \(2^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(0.349154\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5476,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.405051364\)
\(L(\frac12)\) \(\approx\) \(1.405051364\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
37$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 5 T + 27 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - T + 15 T^{2} - p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + T + 35 T^{2} + p T^{3} + p^{2} T^{4} \)
29$C_4$ \( 1 + 3 T - T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 17 T + 133 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 17 T + 153 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 6 T + 90 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 14 T + 162 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 19 T + 211 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 9 T + 123 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 3 T + 117 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 3 T + 59 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
89$C_4$ \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 190 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−14.55568504382690214010878001481, −14.31756017634910343877571717589, −13.64492861912870423155420813297, −13.30390780605588925601099910529, −12.72771702114412320601169767857, −12.35190252318154107531529859448, −11.48124010159742978757620231313, −11.36027416487763192359374783846, −10.49717959069935234629396118610, −10.18811168024303465092774942559, −9.334702209214140742115570595155, −8.350653004117880787826262769279, −7.86716069247191929120431403684, −6.87100223944678328516993531089, −6.01796754899817048898544080092, −5.97691423654313329530298681114, −5.12685721671521497213432134897, −4.44392201551849548294070730841, −3.04662819818334026192895002349, −2.68128995430614572206899555225, 2.68128995430614572206899555225, 3.04662819818334026192895002349, 4.44392201551849548294070730841, 5.12685721671521497213432134897, 5.97691423654313329530298681114, 6.01796754899817048898544080092, 6.87100223944678328516993531089, 7.86716069247191929120431403684, 8.350653004117880787826262769279, 9.334702209214140742115570595155, 10.18811168024303465092774942559, 10.49717959069935234629396118610, 11.36027416487763192359374783846, 11.48124010159742978757620231313, 12.35190252318154107531529859448, 12.72771702114412320601169767857, 13.30390780605588925601099910529, 13.64492861912870423155420813297, 14.31756017634910343877571717589, 14.55568504382690214010878001481

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