Properties

Label 4-637e2-1.1-c1e2-0-14
Degree $4$
Conductor $405769$
Sign $1$
Analytic cond. $25.8721$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 3-s + 4·4-s + 3·6-s + 3·8-s + 3·9-s + 9·11-s + 4·12-s − 2·13-s + 3·16-s − 6·17-s + 9·18-s + 3·19-s + 27·22-s + 3·24-s + 7·25-s − 6·26-s + 8·27-s − 3·29-s + 6·32-s + 9·33-s − 18·34-s + 12·36-s + 9·38-s − 2·39-s − 9·41-s − 11·43-s + ⋯
L(s)  = 1  + 2.12·2-s + 0.577·3-s + 2·4-s + 1.22·6-s + 1.06·8-s + 9-s + 2.71·11-s + 1.15·12-s − 0.554·13-s + 3/4·16-s − 1.45·17-s + 2.12·18-s + 0.688·19-s + 5.75·22-s + 0.612·24-s + 7/5·25-s − 1.17·26-s + 1.53·27-s − 0.557·29-s + 1.06·32-s + 1.56·33-s − 3.08·34-s + 2·36-s + 1.45·38-s − 0.320·39-s − 1.40·41-s − 1.67·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(405769\)    =    \(7^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(25.8721\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{637} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 405769,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.479457682\)
\(L(\frac12)\) \(\approx\) \(8.479457682\)
\(L(\frac{3}{2})\) 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)$
bad7 \( 1 \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good2$C_2^2$ \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 3 T + 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 71 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 154 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.81465526077469446187485662760, −10.63466422616668553527388451937, −9.693273809155868501962119628376, −9.658634634861334117246218853835, −9.018532995973055306697499621477, −8.738198424382278938789984073035, −8.193318940468638333753638078889, −7.46852369968244475550418428448, −6.88284436342934848642114399837, −6.55734967284248223953985967840, −6.55022966253346916898598805457, −5.65996802633039328994567113757, −5.02394722973682661881662539021, −4.57021706512070215732170847634, −4.37979521354408581843519055815, −3.88683415262428967646661788503, −3.15308784196051491565154291431, −3.09067155721651091731584967940, −1.78462523077700191972086828515, −1.39218443189360347516147319258, 1.39218443189360347516147319258, 1.78462523077700191972086828515, 3.09067155721651091731584967940, 3.15308784196051491565154291431, 3.88683415262428967646661788503, 4.37979521354408581843519055815, 4.57021706512070215732170847634, 5.02394722973682661881662539021, 5.65996802633039328994567113757, 6.55022966253346916898598805457, 6.55734967284248223953985967840, 6.88284436342934848642114399837, 7.46852369968244475550418428448, 8.193318940468638333753638078889, 8.738198424382278938789984073035, 9.018532995973055306697499621477, 9.658634634861334117246218853835, 9.693273809155868501962119628376, 10.63466422616668553527388451937, 10.81465526077469446187485662760

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