Properties

Label 4-560e2-1.1-c1e2-0-30
Degree $4$
Conductor $313600$
Sign $1$
Analytic cond. $19.9954$
Root an. cond. $2.11462$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 6·9-s + 8·19-s − 25-s − 4·29-s + 16·31-s + 12·41-s + 12·45-s − 49-s − 8·59-s − 12·61-s − 24·71-s − 8·79-s + 27·81-s + 20·89-s + 16·95-s − 36·101-s − 28·109-s − 22·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + ⋯
L(s)  = 1  + 0.894·5-s + 2·9-s + 1.83·19-s − 1/5·25-s − 0.742·29-s + 2.87·31-s + 1.87·41-s + 1.78·45-s − 1/7·49-s − 1.04·59-s − 1.53·61-s − 2.84·71-s − 0.900·79-s + 3·81-s + 2.11·89-s + 1.64·95-s − 3.58·101-s − 2.68·109-s − 2·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(313600\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(19.9954\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{560} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 313600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.867457141\)
\(L(\frac12)\) \(\approx\) \(2.867457141\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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

−10.64621296412020896649484374983, −10.55710139719972886206778780483, −10.06291567971881036302637130340, −9.624704648885627346441251697559, −9.355483783050034522504474973207, −9.134949576564553691895301969857, −8.135948951116062401115164912635, −7.83201703740521485878140838163, −7.38009549533829100373763125701, −7.03747731098302302180039020275, −6.27437252788839500920666443081, −6.14924573008747085200813762455, −5.40777674015997082377076974892, −4.90098488757384188230680961111, −4.32812241137330901760803721151, −4.00101149824062534662943114451, −3.02442353022434026574576407813, −2.57548060170950369116985658849, −1.48518091307567065337490613066, −1.21483707228753499924510782515, 1.21483707228753499924510782515, 1.48518091307567065337490613066, 2.57548060170950369116985658849, 3.02442353022434026574576407813, 4.00101149824062534662943114451, 4.32812241137330901760803721151, 4.90098488757384188230680961111, 5.40777674015997082377076974892, 6.14924573008747085200813762455, 6.27437252788839500920666443081, 7.03747731098302302180039020275, 7.38009549533829100373763125701, 7.83201703740521485878140838163, 8.135948951116062401115164912635, 9.134949576564553691895301969857, 9.355483783050034522504474973207, 9.624704648885627346441251697559, 10.06291567971881036302637130340, 10.55710139719972886206778780483, 10.64621296412020896649484374983

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