Properties

Label 2-97461-1.1-c1-0-16
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 3·8-s − 4·11-s − 13-s − 16-s − 17-s + 8·19-s + 4·22-s − 5·25-s + 26-s − 8·29-s + 8·31-s − 5·32-s + 34-s − 12·37-s − 8·38-s + 12·41-s − 4·43-s + 4·44-s − 2·47-s + 5·50-s + 52-s + 10·53-s + 8·58-s + 10·59-s − 8·62-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.06·8-s − 1.20·11-s − 0.277·13-s − 1/4·16-s − 0.242·17-s + 1.83·19-s + 0.852·22-s − 25-s + 0.196·26-s − 1.48·29-s + 1.43·31-s − 0.883·32-s + 0.171·34-s − 1.97·37-s − 1.29·38-s + 1.87·41-s − 0.609·43-s + 0.603·44-s − 0.291·47-s + 0.707·50-s + 0.138·52-s + 1.37·53-s + 1.05·58-s + 1.30·59-s − 1.01·62-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: \(1\)
Selberg data: \((2,\ 97461,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.87194306005156, −13.58189478030857, −13.18741527534835, −12.54944088747760, −12.10336464210944, −11.38839666902589, −11.10167352897526, −10.35926060946519, −9.952114875497428, −9.691834362864411, −9.094251354819219, −8.550018565341553, −8.015939638905197, −7.588865180396041, −7.259861206756946, −6.572862877958209, −5.619480057277940, −5.308007350095732, −4.972481870068780, −4.025418414938554, −3.711933380544892, −2.822737159433130, −2.257625059852536, −1.471702263228152, −0.7174832268398778, 0, 0.7174832268398778, 1.471702263228152, 2.257625059852536, 2.822737159433130, 3.711933380544892, 4.025418414938554, 4.972481870068780, 5.308007350095732, 5.619480057277940, 6.572862877958209, 7.259861206756946, 7.588865180396041, 8.015939638905197, 8.550018565341553, 9.094251354819219, 9.691834362864411, 9.952114875497428, 10.35926060946519, 11.10167352897526, 11.38839666902589, 12.10336464210944, 12.54944088747760, 13.18741527534835, 13.58189478030857, 13.87194306005156

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