Properties

Degree $2$
Conductor $97104$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s − 7-s + 9-s − 4·11-s + 6·13-s − 2·15-s + 4·19-s + 21-s + 8·23-s − 25-s − 27-s + 2·29-s + 4·33-s − 2·35-s + 10·37-s − 6·39-s + 6·41-s + 4·43-s + 2·45-s + 49-s + 6·53-s − 8·55-s − 4·57-s − 4·59-s − 6·61-s − 63-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s − 0.516·15-s + 0.917·19-s + 0.218·21-s + 1.66·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.696·33-s − 0.338·35-s + 1.64·37-s − 0.960·39-s + 0.937·41-s + 0.609·43-s + 0.298·45-s + 1/7·49-s + 0.824·53-s − 1.07·55-s − 0.529·57-s − 0.520·59-s − 0.768·61-s − 0.125·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(97104\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 17^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{97104} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 97104,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.059803777\)
\(L(\frac12)\) \(\approx\) \(3.059803777\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
7 \( 1 + T \)
17 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.61806648822537, −13.24629480014929, −12.96654636545788, −12.47933450321913, −11.72139679362974, −11.18412256808507, −10.89869437780965, −10.36216447749733, −9.931213974317194, −9.276865020416131, −9.008514379989283, −8.300438617186593, −7.623886019272862, −7.298817760866614, −6.442251204030785, −6.119551917746038, −5.646147268824590, −5.228465140357660, −4.559176614733807, −3.926060646873681, −3.063752355934742, −2.781609342551095, −1.908108301287838, −1.098403188398378, −0.6690133974279701, 0.6690133974279701, 1.098403188398378, 1.908108301287838, 2.781609342551095, 3.063752355934742, 3.926060646873681, 4.559176614733807, 5.228465140357660, 5.646147268824590, 6.119551917746038, 6.442251204030785, 7.298817760866614, 7.623886019272862, 8.300438617186593, 9.008514379989283, 9.276865020416131, 9.931213974317194, 10.36216447749733, 10.89869437780965, 11.18412256808507, 11.72139679362974, 12.47933450321913, 12.96654636545788, 13.24629480014929, 13.61806648822537

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