Properties

Label 4-857500-1.1-c1e2-0-2
Degree $4$
Conductor $857500$
Sign $1$
Analytic cond. $54.6749$
Root an. cond. $2.71923$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 7-s + 4·9-s − 6·11-s + 16-s − 6·23-s − 28-s + 6·29-s + 4·36-s − 4·37-s + 2·43-s − 6·44-s + 49-s − 6·53-s − 4·63-s + 64-s + 8·67-s − 18·71-s + 6·77-s − 2·79-s + 7·81-s − 6·92-s − 24·99-s + 24·107-s + 22·109-s − 112-s + 24·113-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.377·7-s + 4/3·9-s − 1.80·11-s + 1/4·16-s − 1.25·23-s − 0.188·28-s + 1.11·29-s + 2/3·36-s − 0.657·37-s + 0.304·43-s − 0.904·44-s + 1/7·49-s − 0.824·53-s − 0.503·63-s + 1/8·64-s + 0.977·67-s − 2.13·71-s + 0.683·77-s − 0.225·79-s + 7/9·81-s − 0.625·92-s − 2.41·99-s + 2.32·107-s + 2.10·109-s − 0.0944·112-s + 2.25·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 857500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 857500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(857500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(54.6749\)
Root analytic conductor: \(2.71923\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{857500} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 857500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.836049899\)
\(L(\frac12)\) \(\approx\) \(1.836049899\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5 \( 1 \)
7$C_1$ \( 1 + T \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 106 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 140 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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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

−8.224908148167364547660793016807, −7.61448595613393286757703217481, −7.34439112106834870971810083750, −7.04440581801966531658283867713, −6.40576593098957435909144341550, −5.97707360436401213740399304111, −5.62721286743517268175596275421, −4.87655970304355007216450131088, −4.61582343222263333455122657825, −4.06261170590926391034271732868, −3.28878086518254268163990412705, −2.95912727420180521746052455197, −2.14919988508205179629237745406, −1.77207983872898332466198564186, −0.62587925868655281423821403195, 0.62587925868655281423821403195, 1.77207983872898332466198564186, 2.14919988508205179629237745406, 2.95912727420180521746052455197, 3.28878086518254268163990412705, 4.06261170590926391034271732868, 4.61582343222263333455122657825, 4.87655970304355007216450131088, 5.62721286743517268175596275421, 5.97707360436401213740399304111, 6.40576593098957435909144341550, 7.04440581801966531658283867713, 7.34439112106834870971810083750, 7.61448595613393286757703217481, 8.224908148167364547660793016807

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