Properties

Label 2-91-1.1-c7-0-3
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 14.8·2-s − 5.24·3-s + 93.5·4-s − 195.·5-s + 78.0·6-s + 343·7-s + 513.·8-s − 2.15e3·9-s + 2.91e3·10-s − 3.18e3·11-s − 490.·12-s + 2.19e3·13-s − 5.10e3·14-s + 1.02e3·15-s − 1.96e4·16-s + 6.85e3·17-s + 3.21e4·18-s − 4.25e4·19-s − 1.82e4·20-s − 1.79e3·21-s + 4.73e4·22-s − 6.56e4·23-s − 2.69e3·24-s − 3.98e4·25-s − 3.26e4·26-s + 2.28e4·27-s + 3.20e4·28-s + ⋯
L(s)  = 1  − 1.31·2-s − 0.112·3-s + 0.730·4-s − 0.700·5-s + 0.147·6-s + 0.377·7-s + 0.354·8-s − 0.987·9-s + 0.921·10-s − 0.721·11-s − 0.0819·12-s + 0.277·13-s − 0.497·14-s + 0.0785·15-s − 1.19·16-s + 0.338·17-s + 1.29·18-s − 1.42·19-s − 0.511·20-s − 0.0424·21-s + 0.948·22-s − 1.12·23-s − 0.0397·24-s − 0.509·25-s − 0.364·26-s + 0.222·27-s + 0.276·28-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: \(0\)
Selberg data: \((2,\ 91,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(0.3957124887\)
\(L(\frac12)\) \(\approx\) \(0.3957124887\)
\(L(\frac{9}{2})\) 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 \( 1 + 14.8T + 128T^{2} \)
3 \( 1 + 5.24T + 2.18e3T^{2} \)
5 \( 1 + 195.T + 7.81e4T^{2} \)
11 \( 1 + 3.18e3T + 1.94e7T^{2} \)
17 \( 1 - 6.85e3T + 4.10e8T^{2} \)
19 \( 1 + 4.25e4T + 8.93e8T^{2} \)
23 \( 1 + 6.56e4T + 3.40e9T^{2} \)
29 \( 1 - 6.47e4T + 1.72e10T^{2} \)
31 \( 1 + 3.14e5T + 2.75e10T^{2} \)
37 \( 1 - 4.09e5T + 9.49e10T^{2} \)
41 \( 1 - 2.12e5T + 1.94e11T^{2} \)
43 \( 1 - 7.01e5T + 2.71e11T^{2} \)
47 \( 1 - 1.15e6T + 5.06e11T^{2} \)
53 \( 1 - 2.37e5T + 1.17e12T^{2} \)
59 \( 1 - 1.50e6T + 2.48e12T^{2} \)
61 \( 1 + 2.58e6T + 3.14e12T^{2} \)
67 \( 1 + 3.86e6T + 6.06e12T^{2} \)
71 \( 1 - 2.16e5T + 9.09e12T^{2} \)
73 \( 1 - 4.29e6T + 1.10e13T^{2} \)
79 \( 1 - 6.66e6T + 1.92e13T^{2} \)
83 \( 1 - 1.18e6T + 2.71e13T^{2} \)
89 \( 1 + 9.25e6T + 4.42e13T^{2} \)
97 \( 1 + 4.14e5T + 8.07e13T^{2} \)
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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

−12.38433485240147715876602939794, −11.16495171547365674912320389288, −10.58290629515415408791440911313, −9.173073592181881468474972548787, −8.218761960684568642379118920083, −7.57093612098084437506713269942, −5.86224582454841038259526513759, −4.17837614689931972531957025067, −2.23568639822199071509877170261, −0.47044971842319153896974196468, 0.47044971842319153896974196468, 2.23568639822199071509877170261, 4.17837614689931972531957025067, 5.86224582454841038259526513759, 7.57093612098084437506713269942, 8.218761960684568642379118920083, 9.173073592181881468474972548787, 10.58290629515415408791440911313, 11.16495171547365674912320389288, 12.38433485240147715876602939794

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