Properties

Label 4-2475e2-1.1-c1e2-0-19
Degree $4$
Conductor $6125625$
Sign $1$
Analytic cond. $390.575$
Root an. cond. $4.44555$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 4·7-s + 2·11-s − 4·13-s − 3·16-s + 4·19-s + 4·28-s − 8·31-s − 4·37-s − 4·43-s − 2·44-s − 2·49-s + 4·52-s − 12·53-s + 4·61-s + 7·64-s − 16·67-s − 4·73-s − 4·76-s − 8·77-s − 20·79-s + 24·83-s + 12·89-s + 16·91-s + 20·97-s − 16·103-s + 24·107-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.51·7-s + 0.603·11-s − 1.10·13-s − 3/4·16-s + 0.917·19-s + 0.755·28-s − 1.43·31-s − 0.657·37-s − 0.609·43-s − 0.301·44-s − 2/7·49-s + 0.554·52-s − 1.64·53-s + 0.512·61-s + 7/8·64-s − 1.95·67-s − 0.468·73-s − 0.458·76-s − 0.911·77-s − 2.25·79-s + 2.63·83-s + 1.27·89-s + 1.67·91-s + 2.03·97-s − 1.57·103-s + 2.32·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6125625\)    =    \(3^{4} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(390.575\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2475} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 6125625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 24 T + 298 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−8.814127321635745912038140176294, −8.607522049627000704940417341322, −7.69735998848652636947937930134, −7.66149334077370576666142539244, −7.28546833628206220907980490811, −6.60398367550682932116583418422, −6.54973818543039877891372524390, −6.21411664904222640178475119308, −5.37418924240577003230373375310, −5.37114300791228383583758529819, −4.67505474537592244856044624651, −4.46344441284911895297876364495, −3.71454119464702276352815475694, −3.44657967038959692208149858925, −3.07618491095699059708805535565, −2.48829905494733104436189766369, −1.88100372135999436373448821297, −1.21743386798162512038033609400, 0, 0, 1.21743386798162512038033609400, 1.88100372135999436373448821297, 2.48829905494733104436189766369, 3.07618491095699059708805535565, 3.44657967038959692208149858925, 3.71454119464702276352815475694, 4.46344441284911895297876364495, 4.67505474537592244856044624651, 5.37114300791228383583758529819, 5.37418924240577003230373375310, 6.21411664904222640178475119308, 6.54973818543039877891372524390, 6.60398367550682932116583418422, 7.28546833628206220907980490811, 7.66149334077370576666142539244, 7.69735998848652636947937930134, 8.607522049627000704940417341322, 8.814127321635745912038140176294

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