Properties

Label 4-2646e2-1.1-c1e2-0-34
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 $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3·5-s + 8-s − 3·10-s − 3·11-s + 13-s − 16-s − 6·17-s − 14·19-s + 3·22-s − 9·23-s + 5·25-s − 26-s + 3·29-s − 8·31-s + 6·34-s − 2·37-s + 14·38-s + 3·40-s + 3·41-s + 43-s + 9·46-s − 5·50-s − 6·53-s − 9·55-s − 3·58-s − 2·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.34·5-s + 0.353·8-s − 0.948·10-s − 0.904·11-s + 0.277·13-s − 1/4·16-s − 1.45·17-s − 3.21·19-s + 0.639·22-s − 1.87·23-s + 25-s − 0.196·26-s + 0.557·29-s − 1.43·31-s + 1.02·34-s − 0.328·37-s + 2.27·38-s + 0.474·40-s + 0.468·41-s + 0.152·43-s + 1.32·46-s − 0.707·50-s − 0.824·53-s − 1.21·55-s − 0.393·58-s − 0.256·61-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: \(2\)
Selberg data: \((4,\ 7001316,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + T + T^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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$C_2^2$ \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \)
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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

−8.570193698778401875116366566303, −8.525981555891394904671065376501, −7.914764383414332901513252107540, −7.85093026122805071149489498165, −6.97984422771655409719216593360, −6.77092911531093936737210179287, −6.38488890370501899534161580181, −5.97247407287435424106332010856, −5.80325274642630787284168306162, −5.18493081986776629969099621857, −4.70481232263672350838600291987, −4.27223890798163676500384633114, −4.03727091623910549800775748103, −3.34454478419446373355140836559, −2.47663939708395164456329372230, −2.15025316501747392214106086240, −2.07654012532843986372758369815, −1.35895503129630697372067802881, 0, 0, 1.35895503129630697372067802881, 2.07654012532843986372758369815, 2.15025316501747392214106086240, 2.47663939708395164456329372230, 3.34454478419446373355140836559, 4.03727091623910549800775748103, 4.27223890798163676500384633114, 4.70481232263672350838600291987, 5.18493081986776629969099621857, 5.80325274642630787284168306162, 5.97247407287435424106332010856, 6.38488890370501899534161580181, 6.77092911531093936737210179287, 6.97984422771655409719216593360, 7.85093026122805071149489498165, 7.914764383414332901513252107540, 8.525981555891394904671065376501, 8.570193698778401875116366566303

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