Properties

Label 4-98e2-1.1-c3e2-0-3
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 − 5·3-s − 9·5-s + 10·6-s + 8·8-s + 27·9-s + 18·10-s + 57·11-s + 140·13-s + 45·15-s − 16·16-s + 51·17-s − 54·18-s + 5·19-s − 114·22-s − 69·23-s − 40·24-s + 125·25-s − 280·26-s − 280·27-s + 228·29-s − 90·30-s + 23·31-s − 285·33-s − 102·34-s + 253·37-s − 10·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.962·3-s − 0.804·5-s + 0.680·6-s + 0.353·8-s + 9-s + 0.569·10-s + 1.56·11-s + 2.98·13-s + 0.774·15-s − 1/4·16-s + 0.727·17-s − 0.707·18-s + 0.0603·19-s − 1.10·22-s − 0.625·23-s − 0.340·24-s + 25-s − 2.11·26-s − 1.99·27-s + 1.45·29-s − 0.547·30-s + 0.133·31-s − 1.50·33-s − 0.514·34-s + 1.12·37-s − 0.0426·38-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: induced by $\chi_{98} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9604,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.110007038\)
\(L(\frac12)\) \(\approx\) \(1.110007038\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 + 5 T - 2 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 + 9 T - 44 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 57 T + 1918 T^{2} - 57 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 70 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 3 p T - 8 p^{2} T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 5 T - 6834 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 3 p T - 14 p^{2} T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 114 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 23 T - 29262 T^{2} - 23 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 253 T + 13356 T^{2} - 253 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 42 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 124 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 201 T - 63422 T^{2} - 201 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 393 T + 5572 T^{2} - 393 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 219 T - 157418 T^{2} - 219 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 709 T + 275700 T^{2} + 709 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 419 T - 125202 T^{2} + 419 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 96 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 313 T - 291048 T^{2} + 313 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 461 T - 280518 T^{2} + 461 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 - 588 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 1017 T + 329320 T^{2} + 1017 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 1834 T + p^{3} T^{2} )^{2} \)
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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

−13.62372981054648011903105589250, −13.16259342003854485871889582038, −12.49072710563623397200206454905, −11.81024008987487486658733642658, −11.53567716790889727014184431275, −11.18782076071581031500476251414, −10.38126970777258602197910541954, −10.19176681065728094249705888841, −9.068772783441522075679089040429, −8.946764877558296449285425864421, −8.181234144018956430811129151642, −7.65084223796769923537305049754, −6.83363429475851773424809129261, −6.13690516830978083856269375870, −5.96103970597982696455029135360, −4.65532943160658943114519691213, −3.95753299742786935718457781720, −3.52422175297173536092827535390, −1.37839869608197073295916704853, −0.887277841584653983046864472412, 0.887277841584653983046864472412, 1.37839869608197073295916704853, 3.52422175297173536092827535390, 3.95753299742786935718457781720, 4.65532943160658943114519691213, 5.96103970597982696455029135360, 6.13690516830978083856269375870, 6.83363429475851773424809129261, 7.65084223796769923537305049754, 8.181234144018956430811129151642, 8.946764877558296449285425864421, 9.068772783441522075679089040429, 10.19176681065728094249705888841, 10.38126970777258602197910541954, 11.18782076071581031500476251414, 11.53567716790889727014184431275, 11.81024008987487486658733642658, 12.49072710563623397200206454905, 13.16259342003854485871889582038, 13.62372981054648011903105589250

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