Properties

Label 4-2646e2-1.1-c1e2-0-3
Degree $4$
Conductor $7001316$
Sign $1$
Analytic cond. $446.409$
Root an. cond. $4.59656$
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  + 2-s − 8-s − 3·11-s + 2·13-s − 16-s − 6·17-s + 2·19-s − 3·22-s − 6·23-s + 5·25-s + 2·26-s + 6·29-s − 4·31-s − 6·34-s − 8·37-s + 2·38-s − 9·41-s + 43-s − 6·46-s + 6·47-s + 5·50-s − 24·53-s + 6·58-s − 3·59-s + 8·61-s − 4·62-s + 64-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.353·8-s − 0.904·11-s + 0.554·13-s − 1/4·16-s − 1.45·17-s + 0.458·19-s − 0.639·22-s − 1.25·23-s + 25-s + 0.392·26-s + 1.11·29-s − 0.718·31-s − 1.02·34-s − 1.31·37-s + 0.324·38-s − 1.40·41-s + 0.152·43-s − 0.884·46-s + 0.875·47-s + 0.707·50-s − 3.29·53-s + 0.787·58-s − 0.390·59-s + 1.02·61-s − 0.508·62-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7001316\)    =    \(2^{2} \cdot 3^{6} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(446.409\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7001316,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.215322309\)
\(L(\frac12)\) \(\approx\) \(1.215322309\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−8.951195455316819212044971029963, −8.612558547101274508130640968956, −8.394717026374905204919330072588, −7.86629374566532664746406148414, −7.62011278338385315836029008541, −6.90410473437569512281044297400, −6.80722830266698653883771624004, −6.25848734176310266458490871288, −6.05333590192550963570620669120, −5.38379317077401304612117320241, −5.16101402656647494875328708247, −4.66387952492804353224716605840, −4.48629467992474528871589317040, −3.76138146126832747039547160628, −3.51104101877682604031047978359, −2.94310214646992234496914189736, −2.53097176985130064249099489562, −1.91154260107663790826253474224, −1.39459458437208574765505031194, −0.31277264639312575413714662516, 0.31277264639312575413714662516, 1.39459458437208574765505031194, 1.91154260107663790826253474224, 2.53097176985130064249099489562, 2.94310214646992234496914189736, 3.51104101877682604031047978359, 3.76138146126832747039547160628, 4.48629467992474528871589317040, 4.66387952492804353224716605840, 5.16101402656647494875328708247, 5.38379317077401304612117320241, 6.05333590192550963570620669120, 6.25848734176310266458490871288, 6.80722830266698653883771624004, 6.90410473437569512281044297400, 7.62011278338385315836029008541, 7.86629374566532664746406148414, 8.394717026374905204919330072588, 8.612558547101274508130640968956, 8.951195455316819212044971029963

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