Properties

Label 2-97461-1.1-c1-0-6
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 + 2·5-s − 3·8-s + 2·10-s − 4·11-s − 13-s − 16-s + 17-s − 4·19-s − 2·20-s − 4·22-s + 8·23-s − 25-s − 26-s + 6·29-s + 8·31-s + 5·32-s + 34-s + 2·37-s − 4·38-s − 6·40-s + 6·41-s − 4·43-s + 4·44-s + 8·46-s − 12·47-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.894·5-s − 1.06·8-s + 0.632·10-s − 1.20·11-s − 0.277·13-s − 1/4·16-s + 0.242·17-s − 0.917·19-s − 0.447·20-s − 0.852·22-s + 1.66·23-s − 1/5·25-s − 0.196·26-s + 1.11·29-s + 1.43·31-s + 0.883·32-s + 0.171·34-s + 0.328·37-s − 0.648·38-s − 0.948·40-s + 0.937·41-s − 0.609·43-s + 0.603·44-s + 1.17·46-s − 1.75·47-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

\(L(1)\) \(\approx\) \(2.657891327\)
\(L(\frac12)\) \(\approx\) \(2.657891327\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.70683586662198, −13.26758995506289, −12.94393046676370, −12.54871870962456, −12.05217565440257, −11.24102914716482, −10.96242826109749, −10.12623642384454, −9.883500230300398, −9.490796898218171, −8.754655811813830, −8.234355112011898, −7.998659572240673, −7.006191546438477, −6.513812665644766, −6.106583489587814, −5.381925761673884, −5.037807943565766, −4.672920287137615, −3.975848011099709, −3.173683677691274, −2.698617447990898, −2.273646546955010, −1.242948669384889, −0.4781252724025743, 0.4781252724025743, 1.242948669384889, 2.273646546955010, 2.698617447990898, 3.173683677691274, 3.975848011099709, 4.672920287137615, 5.037807943565766, 5.381925761673884, 6.106583489587814, 6.513812665644766, 7.006191546438477, 7.998659572240673, 8.234355112011898, 8.754655811813830, 9.490796898218171, 9.883500230300398, 10.12623642384454, 10.96242826109749, 11.24102914716482, 12.05217565440257, 12.54871870962456, 12.94393046676370, 13.26758995506289, 13.70683586662198

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