Properties

Label 2-91-1.1-c1-0-6
Degree $2$
Conductor $91$
Sign $-1$
Analytic cond. $0.726638$
Root an. cond. $0.852431$
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  − 2·3-s − 2·4-s − 3·5-s + 7-s + 9-s + 4·12-s + 13-s + 6·15-s + 4·16-s − 6·17-s − 7·19-s + 6·20-s − 2·21-s + 3·23-s + 4·25-s + 4·27-s − 2·28-s − 9·29-s + 5·31-s − 3·35-s − 2·36-s + 2·37-s − 2·39-s − 6·41-s − 43-s − 3·45-s + 3·47-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s − 1.34·5-s + 0.377·7-s + 1/3·9-s + 1.15·12-s + 0.277·13-s + 1.54·15-s + 16-s − 1.45·17-s − 1.60·19-s + 1.34·20-s − 0.436·21-s + 0.625·23-s + 4/5·25-s + 0.769·27-s − 0.377·28-s − 1.67·29-s + 0.898·31-s − 0.507·35-s − 1/3·36-s + 0.328·37-s − 0.320·39-s − 0.937·41-s − 0.152·43-s − 0.447·45-s + 0.437·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{91} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -1)\)

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)$
bad7 \( 1 - T \)
13 \( 1 - T \)
good2 \( 1 + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−13.32859477024006795390919405039, −12.40148043503926800726298674034, −11.34343268397352416353725012574, −10.73267410274711374905403265951, −8.965474064429713311069674592028, −8.023034081311283559976527256533, −6.48432181419363077598714474936, −4.95454046107923865998450494116, −4.04609643315305459582542236808, 0, 4.04609643315305459582542236808, 4.95454046107923865998450494116, 6.48432181419363077598714474936, 8.023034081311283559976527256533, 8.965474064429713311069674592028, 10.73267410274711374905403265951, 11.34343268397352416353725012574, 12.40148043503926800726298674034, 13.32859477024006795390919405039

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