Properties

Label 4-6e8-1.1-c1e2-0-2
Degree $4$
Conductor $1679616$
Sign $1$
Analytic cond. $107.093$
Root an. cond. $3.21692$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s − 4·7-s + 13-s − 6·17-s + 8·19-s + 5·25-s − 9·29-s − 4·31-s − 12·35-s − 2·37-s − 6·41-s + 8·43-s − 12·47-s + 7·49-s − 12·53-s + 61-s + 3·65-s − 4·67-s + 24·71-s + 22·73-s − 16·79-s − 12·83-s − 18·85-s − 6·89-s − 4·91-s + 24·95-s − 2·97-s + ⋯
L(s)  = 1  + 1.34·5-s − 1.51·7-s + 0.277·13-s − 1.45·17-s + 1.83·19-s + 25-s − 1.67·29-s − 0.718·31-s − 2.02·35-s − 0.328·37-s − 0.937·41-s + 1.21·43-s − 1.75·47-s + 49-s − 1.64·53-s + 0.128·61-s + 0.372·65-s − 0.488·67-s + 2.84·71-s + 2.57·73-s − 1.80·79-s − 1.31·83-s − 1.95·85-s − 0.635·89-s − 0.419·91-s + 2.46·95-s − 0.203·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1679616\)    =    \(2^{8} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(107.093\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1679616,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.696728008\)
\(L(\frac12)\) \(\approx\) \(1.696728008\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−9.740973106929054071644412810753, −9.583163462383997357265448249532, −9.363472196036309114401155880280, −8.606098134820989050472078579882, −8.532078949096744569256412001814, −7.72472787141542376709368112231, −7.28639871551875529939875308447, −6.85133063568742233099164183261, −6.61969604341525742722056257545, −5.99570089405431487669368317876, −5.90281503722265118375604612562, −5.18485958916772838256201943208, −5.05918822132392831946448573044, −4.21955633764197007030607844598, −3.61578157063667173648664509794, −3.23201982045174076559310766291, −2.79705550073498154863023221466, −1.96801266879127461765741723686, −1.68915131014969085525096244647, −0.52440637179999905284912982490, 0.52440637179999905284912982490, 1.68915131014969085525096244647, 1.96801266879127461765741723686, 2.79705550073498154863023221466, 3.23201982045174076559310766291, 3.61578157063667173648664509794, 4.21955633764197007030607844598, 5.05918822132392831946448573044, 5.18485958916772838256201943208, 5.90281503722265118375604612562, 5.99570089405431487669368317876, 6.61969604341525742722056257545, 6.85133063568742233099164183261, 7.28639871551875529939875308447, 7.72472787141542376709368112231, 8.532078949096744569256412001814, 8.606098134820989050472078579882, 9.363472196036309114401155880280, 9.583163462383997357265448249532, 9.740973106929054071644412810753

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