Properties

Label 4-4608e2-1.1-c1e2-0-50
Degree $4$
Conductor $21233664$
Sign $1$
Analytic cond. $1353.87$
Root an. cond. $6.06589$
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  − 4·5-s − 4·7-s + 4·11-s + 4·17-s + 8·23-s + 4·25-s − 4·29-s − 12·31-s + 16·35-s − 8·37-s + 12·41-s + 8·43-s + 8·47-s − 4·53-s − 16·55-s − 8·59-s − 8·61-s + 16·67-s + 24·71-s + 8·73-s − 16·77-s − 20·79-s − 4·83-s − 16·85-s − 4·89-s + 4·97-s + 4·101-s + ⋯
L(s)  = 1  − 1.78·5-s − 1.51·7-s + 1.20·11-s + 0.970·17-s + 1.66·23-s + 4/5·25-s − 0.742·29-s − 2.15·31-s + 2.70·35-s − 1.31·37-s + 1.87·41-s + 1.21·43-s + 1.16·47-s − 0.549·53-s − 2.15·55-s − 1.04·59-s − 1.02·61-s + 1.95·67-s + 2.84·71-s + 0.936·73-s − 1.82·77-s − 2.25·79-s − 0.439·83-s − 1.73·85-s − 0.423·89-s + 0.406·97-s + 0.398·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21233664\)    =    \(2^{18} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1353.87\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4608} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 21233664,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2 \( 1 \)
3 \( 1 \)
good5$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
17$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 12 T + 96 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_4$ \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 8 T + 106 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 24 T + 278 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 130 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 20 T + 240 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−7.88167858710803595727087085595, −7.82558676650077260235677786110, −7.29654440008988167172121262173, −7.22745404745803242089590525532, −6.76125500524777023911826131943, −6.43581499984762172646290301150, −5.95494801008077829095745797460, −5.67642853441712098474378789026, −5.02831703732541069481056573312, −4.92519029247021365464585717933, −3.98860244767094249152022917355, −3.87363686950431121127608338070, −3.55820281626850075455067063818, −3.55151630954207785930339717133, −2.60043140340997059868422639638, −2.53779806018088467459933808297, −1.29473671637629931787079199586, −1.17814215515877463189567270695, 0, 0, 1.17814215515877463189567270695, 1.29473671637629931787079199586, 2.53779806018088467459933808297, 2.60043140340997059868422639638, 3.55151630954207785930339717133, 3.55820281626850075455067063818, 3.87363686950431121127608338070, 3.98860244767094249152022917355, 4.92519029247021365464585717933, 5.02831703732541069481056573312, 5.67642853441712098474378789026, 5.95494801008077829095745797460, 6.43581499984762172646290301150, 6.76125500524777023911826131943, 7.22745404745803242089590525532, 7.29654440008988167172121262173, 7.82558676650077260235677786110, 7.88167858710803595727087085595

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