Properties

Label 4-2646e2-1.1-c1e2-0-30
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 − 6·5-s − 4·8-s + 12·10-s − 2·11-s + 5·16-s + 6·19-s − 18·20-s + 4·22-s + 2·23-s + 19·25-s + 8·29-s + 6·31-s − 6·32-s + 2·37-s − 12·38-s + 24·40-s − 18·41-s + 4·43-s − 6·44-s − 4·46-s − 12·47-s − 38·50-s + 16·53-s + 12·55-s − 16·58-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 2.68·5-s − 1.41·8-s + 3.79·10-s − 0.603·11-s + 5/4·16-s + 1.37·19-s − 4.02·20-s + 0.852·22-s + 0.417·23-s + 19/5·25-s + 1.48·29-s + 1.07·31-s − 1.06·32-s + 0.328·37-s − 1.94·38-s + 3.79·40-s − 2.81·41-s + 0.609·43-s − 0.904·44-s − 0.589·46-s − 1.75·47-s − 5.37·50-s + 2.19·53-s + 1.61·55-s − 2.10·58-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: induced by $\chi_{2646} (1, \cdot )$
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$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 6 T + 39 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T + 29 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 8 T + 56 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 2 T + 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 18 T + 155 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 24 T + 258 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 12 T + 152 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 6 T + 133 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$C_4$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 12 T + 104 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \)
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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

−8.407402061190923619451755323100, −8.286792238368949617331425222827, −7.962035694241662108664010302925, −7.70760115156622673825612550338, −7.34433610062321642573310067817, −7.03639357661927983510146439714, −6.53300645337732712080617026304, −6.36881981196400468140037820839, −5.50106927532037167917415566171, −5.12045544713022865077138295735, −4.60330205049140923825091341255, −4.35365403185281411574788853838, −3.60216949801363332690105970281, −3.32796979722954235934600701890, −2.90954569201735560829534542417, −2.54995284564172312716117646968, −1.35050129198020884482127952700, −1.12483444901762346494863767768, 0, 0, 1.12483444901762346494863767768, 1.35050129198020884482127952700, 2.54995284564172312716117646968, 2.90954569201735560829534542417, 3.32796979722954235934600701890, 3.60216949801363332690105970281, 4.35365403185281411574788853838, 4.60330205049140923825091341255, 5.12045544713022865077138295735, 5.50106927532037167917415566171, 6.36881981196400468140037820839, 6.53300645337732712080617026304, 7.03639357661927983510146439714, 7.34433610062321642573310067817, 7.70760115156622673825612550338, 7.962035694241662108664010302925, 8.286792238368949617331425222827, 8.407402061190923619451755323100

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