Properties

Label 4-8092e2-1.1-c1e2-0-2
Degree $4$
Conductor $65480464$
Sign $1$
Analytic cond. $4175.09$
Root an. cond. $8.03834$
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  + 3-s − 5-s + 2·7-s − 2·9-s − 8·11-s − 2·13-s − 15-s + 10·19-s + 2·21-s − 8·23-s − 6·25-s − 2·27-s − 4·29-s − 11·31-s − 8·33-s − 2·35-s + 2·37-s − 2·39-s + 5·41-s − 5·43-s + 2·45-s + 2·47-s + 3·49-s + 5·53-s + 8·55-s + 10·57-s − 16·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.755·7-s − 2/3·9-s − 2.41·11-s − 0.554·13-s − 0.258·15-s + 2.29·19-s + 0.436·21-s − 1.66·23-s − 6/5·25-s − 0.384·27-s − 0.742·29-s − 1.97·31-s − 1.39·33-s − 0.338·35-s + 0.328·37-s − 0.320·39-s + 0.780·41-s − 0.762·43-s + 0.298·45-s + 0.291·47-s + 3/7·49-s + 0.686·53-s + 1.07·55-s + 1.32·57-s − 2.08·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65480464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65480464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(65480464\)    =    \(2^{4} \cdot 7^{2} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(4175.09\)
Root analytic conductor: \(8.03834\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 65480464,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
17 \( 1 \)
good3$D_{4}$ \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 11 T + 89 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 2 T - 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 5 T + 7 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$C_4$ \( 1 + 5 T + 89 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 5 T + 109 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 13 T + 83 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 9 T + 151 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 14 T + 178 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 3 T + 67 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 2 T + 146 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 23 T + 323 T^{2} - 23 p T^{3} + p^{2} T^{4} \)
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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

−7.60618107285647675834737786396, −7.50957415091721636415570775408, −7.34780139711844026350938227739, −6.77782931516280424447754187129, −5.96625102963587290122087145396, −5.85551532866937145647957735204, −5.44164579891992361403282262686, −5.42265490053098422843818840970, −4.71231873130702618614145652308, −4.70849566117855434648787915989, −4.00087020887713892087136578552, −3.60714235422820093751129655167, −3.15546396865274335454400159726, −3.06367076501529526924961044075, −2.29518411667759647948375948098, −2.17232126102681910679320784911, −1.76705197247416741712539884751, −0.935311411160485067139028515470, 0, 0, 0.935311411160485067139028515470, 1.76705197247416741712539884751, 2.17232126102681910679320784911, 2.29518411667759647948375948098, 3.06367076501529526924961044075, 3.15546396865274335454400159726, 3.60714235422820093751129655167, 4.00087020887713892087136578552, 4.70849566117855434648787915989, 4.71231873130702618614145652308, 5.42265490053098422843818840970, 5.44164579891992361403282262686, 5.85551532866937145647957735204, 5.96625102963587290122087145396, 6.77782931516280424447754187129, 7.34780139711844026350938227739, 7.50957415091721636415570775408, 7.60618107285647675834737786396

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