Properties

Label 4-2646e2-1.1-c1e2-0-21
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 8-s + 6·11-s + 2·13-s − 16-s + 3·17-s − 19-s + 6·22-s + 12·23-s − 10·25-s + 2·26-s + 6·29-s − 4·31-s + 3·34-s + 4·37-s − 38-s − 9·41-s + 43-s + 12·46-s + 6·47-s − 10·50-s + 12·53-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 + 1.80·11-s + 0.554·13-s − 1/4·16-s + 0.727·17-s − 0.229·19-s + 1.27·22-s + 2.50·23-s − 2·25-s + 0.392·26-s + 1.11·29-s − 0.718·31-s + 0.514·34-s + 0.657·37-s − 0.162·38-s − 1.40·41-s + 0.152·43-s + 1.76·46-s + 0.875·47-s − 1.41·50-s + 1.64·53-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

\(L(1)\) \(\approx\) \(4.861289238\)
\(L(\frac12)\) \(\approx\) \(4.861289238\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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$C_2^2$ \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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$C_2^2$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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$C_2^2$ \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
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$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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

−8.934572609032773817426846039577, −8.801068843666960516803729481475, −8.297868267920520617097101934547, −8.005442320517725165305069637363, −7.38694678498765606700225856026, −6.99644987445691438954436417764, −6.75496875614406065778576872497, −6.33528618852398443715687928230, −5.99363409826546343911512309911, −5.36060665644419951709524121185, −5.31950809387760137887153122025, −4.70960741806288521832459952492, −4.16514106222346808771996634468, −3.77548366728852827525990345450, −3.66462531008247542517508268212, −3.04244942793068378935527259836, −2.50004754535446845544625641126, −1.84113257286933209092450952087, −1.18747223031186645529145946301, −0.74444858724157812810508037784, 0.74444858724157812810508037784, 1.18747223031186645529145946301, 1.84113257286933209092450952087, 2.50004754535446845544625641126, 3.04244942793068378935527259836, 3.66462531008247542517508268212, 3.77548366728852827525990345450, 4.16514106222346808771996634468, 4.70960741806288521832459952492, 5.31950809387760137887153122025, 5.36060665644419951709524121185, 5.99363409826546343911512309911, 6.33528618852398443715687928230, 6.75496875614406065778576872497, 6.99644987445691438954436417764, 7.38694678498765606700225856026, 8.005442320517725165305069637363, 8.297868267920520617097101934547, 8.801068843666960516803729481475, 8.934572609032773817426846039577

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