Properties

Label 2-97461-1.1-c1-0-17
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  + 2·2-s + 2·4-s + 3·5-s + 6·10-s + 6·11-s + 13-s − 4·16-s − 17-s + 19-s + 6·20-s + 12·22-s + 5·23-s + 4·25-s + 2·26-s + 29-s − 31-s − 8·32-s − 2·34-s + 2·37-s + 2·38-s − 6·41-s − 5·43-s + 12·44-s + 10·46-s + 7·47-s + 8·50-s + 2·52-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 1.34·5-s + 1.89·10-s + 1.80·11-s + 0.277·13-s − 16-s − 0.242·17-s + 0.229·19-s + 1.34·20-s + 2.55·22-s + 1.04·23-s + 4/5·25-s + 0.392·26-s + 0.185·29-s − 0.179·31-s − 1.41·32-s − 0.342·34-s + 0.328·37-s + 0.324·38-s − 0.937·41-s − 0.762·43-s + 1.80·44-s + 1.47·46-s + 1.02·47-s + 1.13·50-s + 0.277·52-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

\(L(1)\) \(\approx\) \(9.489939587\)
\(L(\frac12)\) \(\approx\) \(9.489939587\)
\(L(\frac{3}{2})\) 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 \( 1 - p T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−13.81512884120136, −13.42083828070259, −12.86509096220604, −12.56818215734901, −11.80656713567626, −11.59195727089741, −11.04005944999795, −10.37696982317567, −9.785828464357407, −9.340490791512322, −8.831776509550640, −8.583713128742046, −7.401278030560699, −6.931401402661325, −6.473041050052429, −5.976998205184062, −5.723560924117947, −4.936188121913148, −4.554539530216830, −3.955884756496708, −3.256143328873008, −2.927500154071199, −1.942160536402570, −1.635308555159461, −0.7559223137832555, 0.7559223137832555, 1.635308555159461, 1.942160536402570, 2.927500154071199, 3.256143328873008, 3.955884756496708, 4.554539530216830, 4.936188121913148, 5.723560924117947, 5.976998205184062, 6.473041050052429, 6.931401402661325, 7.401278030560699, 8.583713128742046, 8.831776509550640, 9.340490791512322, 9.785828464357407, 10.37696982317567, 11.04005944999795, 11.59195727089741, 11.80656713567626, 12.56818215734901, 12.86509096220604, 13.42083828070259, 13.81512884120136

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