Properties

Label 4-98e2-1.1-c3e2-0-0
Degree $4$
Conductor $9604$
Sign $1$
Analytic cond. $33.4336$
Root an. cond. $2.40461$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s − 12·5-s + 4·6-s + 8·8-s + 27·9-s + 24·10-s − 48·11-s − 112·13-s + 24·15-s − 16·16-s − 114·17-s − 54·18-s + 2·19-s + 96·22-s + 120·23-s − 16·24-s + 125·25-s + 224·26-s − 154·27-s − 108·29-s − 48·30-s + 236·31-s + 96·33-s + 228·34-s − 146·37-s − 4·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.384·3-s − 1.07·5-s + 0.272·6-s + 0.353·8-s + 9-s + 0.758·10-s − 1.31·11-s − 2.38·13-s + 0.413·15-s − 1/4·16-s − 1.62·17-s − 0.707·18-s + 0.0241·19-s + 0.930·22-s + 1.08·23-s − 0.136·24-s + 25-s + 1.68·26-s − 1.09·27-s − 0.691·29-s − 0.292·30-s + 1.36·31-s + 0.506·33-s + 1.15·34-s − 0.648·37-s − 0.0170·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9604\)    =    \(2^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(33.4336\)
Root analytic conductor: \(2.40461\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9604,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1719135225\)
\(L(\frac12)\) \(\approx\) \(0.1719135225\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p T + p^{2} T^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 + 2 T - 23 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 + 12 T + 19 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 48 T + 973 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 56 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 114 T + 8083 T^{2} + 114 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 2 T - 6855 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 120 T + 2233 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 54 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 236 T + 25905 T^{2} - 236 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 146 T - 29337 T^{2} + 146 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 126 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 376 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 12 T - 103679 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 174 T - 118601 T^{2} + 174 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 138 T - 186335 T^{2} - 138 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 380 T - 82581 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 484 T - 66507 T^{2} - 484 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 576 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 1150 T + 933483 T^{2} + 1150 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 776 T + 109137 T^{2} + 776 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 378 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 390 T - 552869 T^{2} + 390 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 1330 T + p^{3} T^{2} )^{2} \)
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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

−13.27523785637484099221252221109, −13.21432350276452330374150881458, −12.67181885780791612825545727794, −11.94300775304491176090849216710, −11.62377122573260762154199460961, −10.96534257935399045752644406775, −10.42502912410371014268286723448, −9.831290085868944363262197337612, −9.623647697334305041733006581318, −8.517043091481294006301526623494, −8.294477818061506228848922127506, −7.35017591501444721326801329283, −7.19961503421966122236019862978, −6.64398851792384477131846210852, −5.06981349546936126657096608251, −4.98467690433499839281171381842, −4.23762159964574222724258355985, −3.01448533349171404466831915028, −2.01068374645846887997242350460, −0.26356982747853890064697396602, 0.26356982747853890064697396602, 2.01068374645846887997242350460, 3.01448533349171404466831915028, 4.23762159964574222724258355985, 4.98467690433499839281171381842, 5.06981349546936126657096608251, 6.64398851792384477131846210852, 7.19961503421966122236019862978, 7.35017591501444721326801329283, 8.294477818061506228848922127506, 8.517043091481294006301526623494, 9.623647697334305041733006581318, 9.831290085868944363262197337612, 10.42502912410371014268286723448, 10.96534257935399045752644406775, 11.62377122573260762154199460961, 11.94300775304491176090849216710, 12.67181885780791612825545727794, 13.21432350276452330374150881458, 13.27523785637484099221252221109

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