Properties

Label 4-306e2-1.1-c1e2-0-7
Degree $4$
Conductor $93636$
Sign $1$
Analytic cond. $5.97031$
Root an. cond. $1.56314$
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  + 2·2-s + 3·4-s + 4·8-s + 4·13-s + 5·16-s + 6·17-s − 8·19-s + 2·25-s + 8·26-s + 6·32-s + 12·34-s − 16·38-s − 8·43-s + 14·49-s + 4·50-s + 12·52-s − 12·53-s − 24·59-s + 7·64-s − 8·67-s + 18·68-s − 24·76-s + 24·83-s − 16·86-s − 12·89-s + 28·98-s + 6·100-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s + 1.10·13-s + 5/4·16-s + 1.45·17-s − 1.83·19-s + 2/5·25-s + 1.56·26-s + 1.06·32-s + 2.05·34-s − 2.59·38-s − 1.21·43-s + 2·49-s + 0.565·50-s + 1.66·52-s − 1.64·53-s − 3.12·59-s + 7/8·64-s − 0.977·67-s + 2.18·68-s − 2.75·76-s + 2.63·83-s − 1.72·86-s − 1.27·89-s + 2.82·98-s + 3/5·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93636 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93636 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(93636\)    =    \(2^{2} \cdot 3^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(5.97031\)
Root analytic conductor: \(1.56314\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{306} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 93636,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.827774313\)
\(L(\frac12)\) \(\approx\) \(3.827774313\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
17$C_2$ \( 1 - 6 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−11.93607501792769182668295031318, −11.77278203391953279524091178602, −10.89693382820758112729650844712, −10.75233353396644114749424326211, −10.42142763501684524597925503205, −9.720323945926297846589606220523, −9.053198130282691938059944934421, −8.586143756780708080869323228271, −7.940679935227730533037020557938, −7.62751826910890796606047095301, −6.85962790420539338101188665433, −6.35532558205704067565204749071, −5.98640210181096258611109775531, −5.53677329077078390105550412716, −4.62963716484723136444985201560, −4.48036019142334308243976671589, −3.41834532574028591770644406293, −3.35726652086495229374274367534, −2.26836199110576836875905430089, −1.40822723952589668292103229810, 1.40822723952589668292103229810, 2.26836199110576836875905430089, 3.35726652086495229374274367534, 3.41834532574028591770644406293, 4.48036019142334308243976671589, 4.62963716484723136444985201560, 5.53677329077078390105550412716, 5.98640210181096258611109775531, 6.35532558205704067565204749071, 6.85962790420539338101188665433, 7.62751826910890796606047095301, 7.940679935227730533037020557938, 8.586143756780708080869323228271, 9.053198130282691938059944934421, 9.720323945926297846589606220523, 10.42142763501684524597925503205, 10.75233353396644114749424326211, 10.89693382820758112729650844712, 11.77278203391953279524091178602, 11.93607501792769182668295031318

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