Properties

Label 2-97461-1.1-c1-0-18
Degree $2$
Conductor $97461$
Sign $1$
Analytic cond. $778.230$
Root an. cond. $27.8967$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 4·5-s − 3·8-s + 4·10-s + 13-s − 16-s + 17-s + 4·19-s − 4·20-s + 2·23-s + 11·25-s + 26-s + 8·31-s + 5·32-s + 34-s + 10·37-s + 4·38-s − 12·40-s + 4·43-s + 2·46-s − 6·47-s + 11·50-s − 52-s + 12·53-s + 6·59-s − 2·61-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 1.78·5-s − 1.06·8-s + 1.26·10-s + 0.277·13-s − 1/4·16-s + 0.242·17-s + 0.917·19-s − 0.894·20-s + 0.417·23-s + 11/5·25-s + 0.196·26-s + 1.43·31-s + 0.883·32-s + 0.171·34-s + 1.64·37-s + 0.648·38-s − 1.89·40-s + 0.609·43-s + 0.294·46-s − 0.875·47-s + 1.55·50-s − 0.138·52-s + 1.64·53-s + 0.781·59-s − 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(97461\)    =    \(3^{2} \cdot 7^{2} \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(778.230\)
Root analytic conductor: \(27.8967\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{97461} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 97461,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.888611471\)
\(L(\frac12)\) \(\approx\) \(5.888611471\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 - T \)
17 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
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

−13.78975435955952, −13.25256604738538, −13.08340204639713, −12.56063655448218, −11.88046176404497, −11.48958414295692, −10.78809579979165, −10.08317279535237, −9.888543728837344, −9.382106721493938, −8.959352571051172, −8.393430998100015, −7.827024878127247, −6.951825398325341, −6.519844270048652, −5.926312025105655, −5.579899539792665, −5.175966993501507, −4.518496478407001, −4.034856307283116, −3.076138284684015, −2.812897391974655, −2.129242905493500, −1.204041444727229, −0.7678379406635504, 0.7678379406635504, 1.204041444727229, 2.129242905493500, 2.812897391974655, 3.076138284684015, 4.034856307283116, 4.518496478407001, 5.175966993501507, 5.579899539792665, 5.926312025105655, 6.519844270048652, 6.951825398325341, 7.827024878127247, 8.393430998100015, 8.959352571051172, 9.382106721493938, 9.888543728837344, 10.08317279535237, 10.78809579979165, 11.48958414295692, 11.88046176404497, 12.56063655448218, 13.08340204639713, 13.25256604738538, 13.78975435955952

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