Properties

Label 4-1575e2-1.1-c1e2-0-0
Degree $4$
Conductor $2480625$
Sign $1$
Analytic cond. $158.166$
Root an. cond. $3.54632$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·4-s − 8·11-s + 5·16-s − 8·19-s − 4·29-s − 4·41-s − 24·44-s − 49-s + 24·59-s − 4·61-s + 3·64-s − 24·76-s + 32·79-s − 28·89-s − 28·101-s + 36·109-s − 12·116-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 12·164-s + ⋯
L(s)  = 1  + 3/2·4-s − 2.41·11-s + 5/4·16-s − 1.83·19-s − 0.742·29-s − 0.624·41-s − 3.61·44-s − 1/7·49-s + 3.12·59-s − 0.512·61-s + 3/8·64-s − 2.75·76-s + 3.60·79-s − 2.96·89-s − 2.78·101-s + 3.44·109-s − 1.11·116-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s − 0.937·164-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2480625\)    =    \(3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(158.166\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1575} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2480625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.736576929\)
\(L(\frac12)\) \(\approx\) \(1.736576929\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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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

−9.846819165895579168648776352891, −9.288907538423299691858418652615, −8.568308852507399541102074920926, −8.290862589208137705549281831045, −8.145397375085014274993227138637, −7.47861666282821249219721904126, −7.32937424660704620468369831124, −6.75686578494796504100488864818, −6.54635102394142019942638922991, −5.88257022590385951517029029387, −5.70371198742171042777019314768, −4.97302012218261490490685459791, −4.96457990718881913342270590498, −3.96798820885129747047588766959, −3.70289713905675732950209001901, −2.73165663838727285489596261049, −2.70845922125454300603172647268, −2.11581889590910888248287043879, −1.71543307717574411863414811790, −0.46116509224226748845528394685, 0.46116509224226748845528394685, 1.71543307717574411863414811790, 2.11581889590910888248287043879, 2.70845922125454300603172647268, 2.73165663838727285489596261049, 3.70289713905675732950209001901, 3.96798820885129747047588766959, 4.96457990718881913342270590498, 4.97302012218261490490685459791, 5.70371198742171042777019314768, 5.88257022590385951517029029387, 6.54635102394142019942638922991, 6.75686578494796504100488864818, 7.32937424660704620468369831124, 7.47861666282821249219721904126, 8.145397375085014274993227138637, 8.290862589208137705549281831045, 8.568308852507399541102074920926, 9.288907538423299691858418652615, 9.846819165895579168648776352891

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