Properties

Label 4-75e2-1.1-c21e2-0-1
Degree $4$
Conductor $5625$
Sign $1$
Analytic cond. $43935.5$
Root an. cond. $14.4778$
Motivic weight $21$
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  − 3.89e6·4-s − 3.48e9·9-s + 2.91e10·11-s + 1.07e13·16-s + 5.84e13·19-s − 4.80e15·29-s + 4.47e15·31-s + 1.35e16·36-s − 2.06e17·41-s − 1.13e17·44-s + 9.85e17·49-s − 1.10e19·59-s − 1.43e19·61-s − 2.47e19·64-s + 5.29e19·71-s − 2.27e20·76-s + 3.37e19·79-s + 1.21e19·81-s + 6.25e20·89-s − 1.01e20·99-s + 2.89e20·101-s − 4.49e20·109-s + 1.86e22·116-s − 1.41e22·121-s − 1.74e22·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1.85·4-s − 1/3·9-s + 0.339·11-s + 2.44·16-s + 2.18·19-s − 2.11·29-s + 0.981·31-s + 0.618·36-s − 2.40·41-s − 0.629·44-s + 1.76·49-s − 2.81·59-s − 2.57·61-s − 2.68·64-s + 1.92·71-s − 4.05·76-s + 0.401·79-s + 1/9·81-s + 2.12·89-s − 0.113·99-s + 0.260·101-s − 0.181·109-s + 3.93·116-s − 1.91·121-s − 1.82·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+21/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5625\)    =    \(3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(43935.5\)
Root analytic conductor: \(14.4778\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{75} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5625,\ (\ :21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(1.230730571\)
\(L(\frac12)\) \(\approx\) \(1.230730571\)
\(L(\frac{23}{2})\) 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$C_2$ \( 1 + p^{20} T^{2} \)
5 \( 1 \)
good2$C_2^2$ \( 1 + 243377 p^{4} T^{2} + p^{42} T^{4} \)
7$C_2^2$ \( 1 - 20104122242432686 p^{2} T^{2} + p^{42} T^{4} \)
11$C_2$ \( ( 1 - 1325621196 p T + p^{21} T^{2} )^{2} \)
13$C_2^2$ \( 1 - \)\(48\!\cdots\!22\)\( T^{2} + p^{42} T^{4} \)
17$C_2^2$ \( 1 - \)\(22\!\cdots\!90\)\( p^{2} T^{2} + p^{42} T^{4} \)
19$C_2$ \( ( 1 - 1536996803884 p T + p^{21} T^{2} )^{2} \)
23$C_2^2$ \( 1 - \)\(54\!\cdots\!46\)\( T^{2} + p^{42} T^{4} \)
29$C_2$ \( ( 1 + 2400788707090758 T + p^{21} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2239820676947000 T + p^{21} T^{2} )^{2} \)
37$C_2^2$ \( 1 - \)\(76\!\cdots\!74\)\( T^{2} + p^{42} T^{4} \)
41$C_2$ \( ( 1 + 103207571041281030 T + p^{21} T^{2} )^{2} \)
43$C_2^2$ \( 1 - \)\(12\!\cdots\!10\)\( T^{2} + p^{42} T^{4} \)
47$C_2^2$ \( 1 - \)\(25\!\cdots\!30\)\( T^{2} + p^{42} T^{4} \)
53$C_2^2$ \( 1 - \)\(30\!\cdots\!06\)\( T^{2} + p^{42} T^{4} \)
59$C_2$ \( ( 1 + 5534365798259081316 T + p^{21} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 7176205164722961202 T + p^{21} T^{2} )^{2} \)
67$C_2^2$ \( 1 - \)\(19\!\cdots\!90\)\( T^{2} + p^{42} T^{4} \)
71$C_2$ \( ( 1 - 26457854874259376232 T + p^{21} T^{2} )^{2} \)
73$C_2^2$ \( 1 - \)\(25\!\cdots\!46\)\( T^{2} + p^{42} T^{4} \)
79$C_2$ \( ( 1 - 16886125085525986840 T + p^{21} T^{2} )^{2} \)
83$C_2^2$ \( 1 - \)\(10\!\cdots\!82\)\( T^{2} + p^{42} T^{4} \)
89$C_2$ \( ( 1 - \)\(31\!\cdots\!86\)\( T + p^{21} T^{2} )^{2} \)
97$C_2^2$ \( 1 - \)\(15\!\cdots\!70\)\( T^{2} + p^{42} T^{4} \)
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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

−10.70395271595365656400159869689, −10.31790327359193048529385548741, −9.540105257839947541989109318210, −9.213803361351557768225144693957, −9.136884268012323542494651105478, −8.203183620132552849886121851833, −7.85707229564663456303443403208, −7.36437350569723422554403866589, −6.55895443587506111554054254622, −5.67961360659597809460663018095, −5.55083429288188830203847058749, −4.78787285958715234041296435553, −4.57260614924339785961083986405, −3.59590640501014609903974231089, −3.49597279815061322255561467576, −2.90913331024900715429975787473, −1.83237557819244857294713399103, −1.35881730184474954745301644058, −0.72897532887133491439856541628, −0.29829578101666617833321259049, 0.29829578101666617833321259049, 0.72897532887133491439856541628, 1.35881730184474954745301644058, 1.83237557819244857294713399103, 2.90913331024900715429975787473, 3.49597279815061322255561467576, 3.59590640501014609903974231089, 4.57260614924339785961083986405, 4.78787285958715234041296435553, 5.55083429288188830203847058749, 5.67961360659597809460663018095, 6.55895443587506111554054254622, 7.36437350569723422554403866589, 7.85707229564663456303443403208, 8.203183620132552849886121851833, 9.136884268012323542494651105478, 9.213803361351557768225144693957, 9.540105257839947541989109318210, 10.31790327359193048529385548741, 10.70395271595365656400159869689

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