Properties

Label 4-350e2-1.1-c1e2-0-11
Degree $4$
Conductor $122500$
Sign $1$
Analytic cond. $7.81070$
Root an. cond. $1.67175$
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  − 4-s + 6·9-s + 8·11-s + 16-s − 12·29-s + 16·31-s − 6·36-s + 4·41-s − 8·44-s − 49-s + 16·59-s − 28·61-s − 64-s − 32·71-s + 16·79-s + 27·81-s − 20·89-s + 48·99-s − 12·101-s − 12·109-s + 12·116-s + 26·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1/2·4-s + 2·9-s + 2.41·11-s + 1/4·16-s − 2.22·29-s + 2.87·31-s − 36-s + 0.624·41-s − 1.20·44-s − 1/7·49-s + 2.08·59-s − 3.58·61-s − 1/8·64-s − 3.79·71-s + 1.80·79-s + 3·81-s − 2.11·89-s + 4.82·99-s − 1.19·101-s − 1.14·109-s + 1.11·116-s + 2.36·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(122500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(7.81070\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{350} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 122500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.965320765\)
\(L(\frac12)\) \(\approx\) \(1.965320765\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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

−11.73524507409310901629681611586, −11.44513264493481635354231646084, −10.65002353250020221104707220904, −10.28837052092234625714461130138, −9.728098846078623639997935622348, −9.494747861583977445836362589452, −9.012486951053894611631767077035, −8.688116991019315822572852464589, −7.67144460496492996069726098658, −7.64306579165661627096830496648, −6.86420041488570947422359610279, −6.46439935707185610272543985490, −6.11189048839258655266713140206, −5.22914717908928257419633570309, −4.48247240717486246287022733395, −4.08160789816250886914506969336, −3.92999144609648339195991388200, −2.89477719381851478730103258526, −1.59233549458569435829285893892, −1.24175101525618634516311581904, 1.24175101525618634516311581904, 1.59233549458569435829285893892, 2.89477719381851478730103258526, 3.92999144609648339195991388200, 4.08160789816250886914506969336, 4.48247240717486246287022733395, 5.22914717908928257419633570309, 6.11189048839258655266713140206, 6.46439935707185610272543985490, 6.86420041488570947422359610279, 7.64306579165661627096830496648, 7.67144460496492996069726098658, 8.688116991019315822572852464589, 9.012486951053894611631767077035, 9.494747861583977445836362589452, 9.728098846078623639997935622348, 10.28837052092234625714461130138, 10.65002353250020221104707220904, 11.44513264493481635354231646084, 11.73524507409310901629681611586

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