Properties

Label 4-74e2-1.1-c1e2-0-1
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 + 3·3-s + 3·4-s − 5-s − 6·6-s + 2·7-s − 4·8-s + 4·9-s + 2·10-s − 11-s + 9·12-s − 13-s − 4·14-s − 3·15-s + 5·16-s − 12·17-s − 8·18-s + 4·19-s − 3·20-s + 6·21-s + 2·22-s − 3·23-s − 12·24-s − 6·25-s + 2·26-s + 6·27-s + 6·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.73·3-s + 3/2·4-s − 0.447·5-s − 2.44·6-s + 0.755·7-s − 1.41·8-s + 4/3·9-s + 0.632·10-s − 0.301·11-s + 2.59·12-s − 0.277·13-s − 1.06·14-s − 0.774·15-s + 5/4·16-s − 2.91·17-s − 1.88·18-s + 0.917·19-s − 0.670·20-s + 1.30·21-s + 0.426·22-s − 0.625·23-s − 2.44·24-s − 6/5·25-s + 0.392·26-s + 1.15·27-s + 1.13·28-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: induced by $\chi_{74} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5476,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7296348316\)
\(L(\frac12)\) \(\approx\) \(0.7296348316\)
\(L(\frac{3}{2})\) 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$C_4$ \( 1 - p T + 5 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + T + 19 T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + T + 23 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_4$ \( 1 - 3 T + 31 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 3 T + 61 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 9 T + 73 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 3 T + 43 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 11 T + 83 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 21 T + 253 T^{2} + 21 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 7 T + 11 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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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

−14.85715955519318499835389416333, −14.49667430882702977079802065413, −13.82810602652582601657229089158, −13.45578658411617519237182903232, −12.72332663292485060593600802018, −11.90105752959434064909667781550, −11.33074162447793271165656847275, −11.00558376992052385392883780627, −10.11301099796250134694050439536, −9.654858831283765204535950149774, −8.841253477501806828214993356905, −8.769483072722598629061956387655, −7.900095471836613411558585840504, −7.86588361361950631323811500192, −6.96682723610503755952759087238, −6.27120580306579210087654420633, −4.84003718833468002241545765906, −3.93060845224348257138198386426, −2.69741759734566796341285138193, −2.09579133783652754251714950349, 2.09579133783652754251714950349, 2.69741759734566796341285138193, 3.93060845224348257138198386426, 4.84003718833468002241545765906, 6.27120580306579210087654420633, 6.96682723610503755952759087238, 7.86588361361950631323811500192, 7.900095471836613411558585840504, 8.769483072722598629061956387655, 8.841253477501806828214993356905, 9.654858831283765204535950149774, 10.11301099796250134694050439536, 11.00558376992052385392883780627, 11.33074162447793271165656847275, 11.90105752959434064909667781550, 12.72332663292485060593600802018, 13.45578658411617519237182903232, 13.82810602652582601657229089158, 14.49667430882702977079802065413, 14.85715955519318499835389416333

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