Properties

Label 4-9576e2-1.1-c1e2-0-15
Degree $4$
Conductor $91699776$
Sign $1$
Analytic cond. $5846.85$
Root an. cond. $8.74441$
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  − 2·5-s + 2·7-s − 4·11-s + 2·17-s − 2·19-s + 4·23-s − 2·25-s − 2·29-s − 12·31-s − 4·35-s + 8·37-s − 8·41-s + 16·43-s − 10·47-s + 3·49-s + 18·53-s + 8·55-s − 12·67-s + 2·71-s + 16·73-s − 8·77-s + 8·79-s − 14·83-s − 4·85-s + 4·95-s − 8·97-s − 26·101-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.755·7-s − 1.20·11-s + 0.485·17-s − 0.458·19-s + 0.834·23-s − 2/5·25-s − 0.371·29-s − 2.15·31-s − 0.676·35-s + 1.31·37-s − 1.24·41-s + 2.43·43-s − 1.45·47-s + 3/7·49-s + 2.47·53-s + 1.07·55-s − 1.46·67-s + 0.237·71-s + 1.87·73-s − 0.911·77-s + 0.900·79-s − 1.53·83-s − 0.433·85-s + 0.410·95-s − 0.812·97-s − 2.58·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(91699776\)    =    \(2^{6} \cdot 3^{4} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(5846.85\)
Root analytic conductor: \(8.74441\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 91699776,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 10 T + 74 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 2 T + 98 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + 14 T + 170 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−7.44501949586773865423740920188, −7.41551329369254784495455900500, −6.92393248167170013205876388661, −6.67391760073726690808913027879, −5.94849896135790945206059990784, −5.80659710904386742594266225058, −5.36617909703670805193324771814, −5.22984499420197939245598446605, −4.70221612849787860038625425424, −4.42663782277204671861934277480, −3.92647562332410808656653822712, −3.69024014555292146124898775045, −3.38149204607146716231062852008, −2.70520558057069912964138036903, −2.36809958452679431388331325289, −2.15370415907958461133147686694, −1.28631455136984362766715061694, −1.09496649740619004624668896937, 0, 0, 1.09496649740619004624668896937, 1.28631455136984362766715061694, 2.15370415907958461133147686694, 2.36809958452679431388331325289, 2.70520558057069912964138036903, 3.38149204607146716231062852008, 3.69024014555292146124898775045, 3.92647562332410808656653822712, 4.42663782277204671861934277480, 4.70221612849787860038625425424, 5.22984499420197939245598446605, 5.36617909703670805193324771814, 5.80659710904386742594266225058, 5.94849896135790945206059990784, 6.67391760073726690808913027879, 6.92393248167170013205876388661, 7.41551329369254784495455900500, 7.44501949586773865423740920188

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