Properties

Label 2-91-1.1-c7-0-29
Degree $2$
Conductor $91$
Sign $-1$
Analytic cond. $28.4270$
Root an. cond. $5.33170$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 13.8·2-s + 79.8·3-s + 64.3·4-s − 355.·5-s − 1.10e3·6-s + 343·7-s + 882.·8-s + 4.18e3·9-s + 4.93e3·10-s − 3.97e3·11-s + 5.13e3·12-s − 2.19e3·13-s − 4.75e3·14-s − 2.83e4·15-s − 2.04e4·16-s + 3.20e4·17-s − 5.80e4·18-s − 3.46e4·19-s − 2.29e4·20-s + 2.73e4·21-s + 5.51e4·22-s − 3.52e4·23-s + 7.04e4·24-s + 4.83e4·25-s + 3.04e4·26-s + 1.59e5·27-s + 2.20e4·28-s + ⋯
L(s)  = 1  − 1.22·2-s + 1.70·3-s + 0.503·4-s − 1.27·5-s − 2.09·6-s + 0.377·7-s + 0.609·8-s + 1.91·9-s + 1.56·10-s − 0.901·11-s + 0.858·12-s − 0.277·13-s − 0.463·14-s − 2.17·15-s − 1.24·16-s + 1.58·17-s − 2.34·18-s − 1.15·19-s − 0.640·20-s + 0.645·21-s + 1.10·22-s − 0.603·23-s + 1.04·24-s + 0.619·25-s + 0.340·26-s + 1.55·27-s + 0.190·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(28.4270\)
Root analytic conductor: \(5.33170\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 91,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - 343T \)
13 \( 1 + 2.19e3T \)
good2 \( 1 + 13.8T + 128T^{2} \)
3 \( 1 - 79.8T + 2.18e3T^{2} \)
5 \( 1 + 355.T + 7.81e4T^{2} \)
11 \( 1 + 3.97e3T + 1.94e7T^{2} \)
17 \( 1 - 3.20e4T + 4.10e8T^{2} \)
19 \( 1 + 3.46e4T + 8.93e8T^{2} \)
23 \( 1 + 3.52e4T + 3.40e9T^{2} \)
29 \( 1 + 1.61e5T + 1.72e10T^{2} \)
31 \( 1 - 3.15e5T + 2.75e10T^{2} \)
37 \( 1 + 4.47e5T + 9.49e10T^{2} \)
41 \( 1 + 5.87e5T + 1.94e11T^{2} \)
43 \( 1 - 4.31e5T + 2.71e11T^{2} \)
47 \( 1 + 1.09e6T + 5.06e11T^{2} \)
53 \( 1 + 1.10e6T + 1.17e12T^{2} \)
59 \( 1 + 2.03e6T + 2.48e12T^{2} \)
61 \( 1 + 1.10e5T + 3.14e12T^{2} \)
67 \( 1 + 1.31e6T + 6.06e12T^{2} \)
71 \( 1 + 3.26e6T + 9.09e12T^{2} \)
73 \( 1 - 4.07e5T + 1.10e13T^{2} \)
79 \( 1 + 4.59e6T + 1.92e13T^{2} \)
83 \( 1 + 2.19e6T + 2.71e13T^{2} \)
89 \( 1 - 8.51e6T + 4.42e13T^{2} \)
97 \( 1 - 8.73e5T + 8.07e13T^{2} \)
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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

−12.08186049586631986906771536207, −10.57313147305697761646588936935, −9.705565609986521738440430165052, −8.407140531253178250913579359742, −8.061092284932148524636953709698, −7.35201737992491333176015863373, −4.49694805464871564065538603051, −3.22282454670445935353522658596, −1.72815867989601915891903134667, 0, 1.72815867989601915891903134667, 3.22282454670445935353522658596, 4.49694805464871564065538603051, 7.35201737992491333176015863373, 8.061092284932148524636953709698, 8.407140531253178250913579359742, 9.705565609986521738440430165052, 10.57313147305697761646588936935, 12.08186049586631986906771536207

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