Properties

Label 4-2646e2-1.1-c1e2-0-35
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s + 2·5-s − 4·8-s − 4·10-s − 2·11-s − 4·13-s + 5·16-s − 2·17-s + 4·19-s + 6·20-s + 4·22-s − 8·23-s + 8·26-s − 10·29-s − 4·31-s − 6·32-s + 4·34-s − 8·37-s − 8·38-s − 8·40-s − 6·41-s + 10·43-s − 6·44-s + 16·46-s + 6·47-s − 12·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.894·5-s − 1.41·8-s − 1.26·10-s − 0.603·11-s − 1.10·13-s + 5/4·16-s − 0.485·17-s + 0.917·19-s + 1.34·20-s + 0.852·22-s − 1.66·23-s + 1.56·26-s − 1.85·29-s − 0.718·31-s − 1.06·32-s + 0.685·34-s − 1.31·37-s − 1.29·38-s − 1.26·40-s − 0.937·41-s + 1.52·43-s − 0.904·44-s + 2.35·46-s + 0.875·47-s − 1.66·52-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$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + 16 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T + 28 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 10 T + 76 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T + 59 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 22 T + 232 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 20 T + 215 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 6 T + 115 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 26 T + 304 T^{2} + 26 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 16 T + 202 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 6 T + 124 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 6 T + 91 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−8.709894022894092850540669708472, −8.399030497591828844802120377317, −7.70796465662326201273858237927, −7.68885320052820091578852436965, −7.31974379342419893642196146749, −7.01226476655276887425478132956, −6.28123610480342635751617364749, −6.16118333744205388885565266414, −5.64354273892781596066807266471, −5.27893624167987078723081192796, −4.98517145628389106561709250795, −4.22513355675809445930178184655, −3.71779790363866877236405743123, −3.23747365183452452773022089499, −2.50766523018405897724107243041, −2.30815047777987556686118546361, −1.73702203452452869574318610693, −1.36405521513631835469785042490, 0, 0, 1.36405521513631835469785042490, 1.73702203452452869574318610693, 2.30815047777987556686118546361, 2.50766523018405897724107243041, 3.23747365183452452773022089499, 3.71779790363866877236405743123, 4.22513355675809445930178184655, 4.98517145628389106561709250795, 5.27893624167987078723081192796, 5.64354273892781596066807266471, 6.16118333744205388885565266414, 6.28123610480342635751617364749, 7.01226476655276887425478132956, 7.31974379342419893642196146749, 7.68885320052820091578852436965, 7.70796465662326201273858237927, 8.399030497591828844802120377317, 8.709894022894092850540669708472

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