Properties

Label 4-637e2-1.1-c1e2-0-21
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 4·4-s − 3·8-s − 9-s − 6·11-s + 2·13-s + 3·16-s + 6·17-s + 3·18-s − 6·19-s + 18·22-s − 12·23-s − 5·25-s − 6·26-s − 10·31-s − 6·32-s − 18·34-s − 4·36-s − 4·37-s + 18·38-s − 16·43-s − 24·44-s + 36·46-s + 6·47-s + 15·50-s + 8·52-s − 6·53-s + ⋯
L(s)  = 1  − 2.12·2-s + 2·4-s − 1.06·8-s − 1/3·9-s − 1.80·11-s + 0.554·13-s + 3/4·16-s + 1.45·17-s + 0.707·18-s − 1.37·19-s + 3.83·22-s − 2.50·23-s − 25-s − 1.17·26-s − 1.79·31-s − 1.06·32-s − 3.08·34-s − 2/3·36-s − 0.657·37-s + 2.91·38-s − 2.43·43-s − 3.61·44-s + 5.30·46-s + 0.875·47-s + 2.12·50-s + 1.10·52-s − 0.824·53-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: \(2\)
Selberg data: \((4,\ 405769,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad7 \( 1 \)
13$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 6 T + 95 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 107 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 8 T + 117 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 173 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−10.26813337319332781669343398813, −9.926967043435727606751488411465, −9.535584811349761561598349820299, −9.073591398621732359811111073913, −8.460241464774015638621155340295, −8.279674204242944344175756530563, −7.979585550272571007696206837437, −7.53959975314164666285963391409, −7.27230104557089325010484293257, −6.20773811119268649499049361066, −6.07649062075562653452378841332, −5.48987176006989565595941322782, −5.02887787297504592985778238416, −4.09084762809047939404241567772, −3.55744827318528851282072200960, −2.90462871799594758035438569900, −1.95910020964794718294263088171, −1.63436554889716791348719925216, 0, 0, 1.63436554889716791348719925216, 1.95910020964794718294263088171, 2.90462871799594758035438569900, 3.55744827318528851282072200960, 4.09084762809047939404241567772, 5.02887787297504592985778238416, 5.48987176006989565595941322782, 6.07649062075562653452378841332, 6.20773811119268649499049361066, 7.27230104557089325010484293257, 7.53959975314164666285963391409, 7.979585550272571007696206837437, 8.279674204242944344175756530563, 8.460241464774015638621155340295, 9.073591398621732359811111073913, 9.535584811349761561598349820299, 9.926967043435727606751488411465, 10.26813337319332781669343398813

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