Properties

Label 2-91-1.1-c7-0-23
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  − 11.5·2-s + 23.9·3-s + 4.51·4-s − 301.·5-s − 275.·6-s + 343·7-s + 1.42e3·8-s − 1.61e3·9-s + 3.46e3·10-s + 8.16e3·11-s + 107.·12-s − 2.19e3·13-s − 3.94e3·14-s − 7.20e3·15-s − 1.69e4·16-s − 1.84e4·17-s + 1.85e4·18-s + 2.54e4·19-s − 1.35e3·20-s + 8.19e3·21-s − 9.39e4·22-s + 1.00e5·23-s + 3.39e4·24-s + 1.27e4·25-s + 2.52e4·26-s − 9.09e4·27-s + 1.54e3·28-s + ⋯
L(s)  = 1  − 1.01·2-s + 0.511·3-s + 0.0352·4-s − 1.07·5-s − 0.520·6-s + 0.377·7-s + 0.981·8-s − 0.738·9-s + 1.09·10-s + 1.84·11-s + 0.0180·12-s − 0.277·13-s − 0.384·14-s − 0.551·15-s − 1.03·16-s − 0.910·17-s + 0.751·18-s + 0.851·19-s − 0.0380·20-s + 0.193·21-s − 1.88·22-s + 1.72·23-s + 0.501·24-s + 0.162·25-s + 0.282·26-s − 0.888·27-s + 0.0133·28-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 \( 1 + 11.5T + 128T^{2} \)
3 \( 1 - 23.9T + 2.18e3T^{2} \)
5 \( 1 + 301.T + 7.81e4T^{2} \)
11 \( 1 - 8.16e3T + 1.94e7T^{2} \)
17 \( 1 + 1.84e4T + 4.10e8T^{2} \)
19 \( 1 - 2.54e4T + 8.93e8T^{2} \)
23 \( 1 - 1.00e5T + 3.40e9T^{2} \)
29 \( 1 - 1.63e3T + 1.72e10T^{2} \)
31 \( 1 + 1.83e5T + 2.75e10T^{2} \)
37 \( 1 - 5.64e5T + 9.49e10T^{2} \)
41 \( 1 + 7.38e5T + 1.94e11T^{2} \)
43 \( 1 + 7.59e5T + 2.71e11T^{2} \)
47 \( 1 + 1.10e6T + 5.06e11T^{2} \)
53 \( 1 + 8.22e5T + 1.17e12T^{2} \)
59 \( 1 + 7.68e5T + 2.48e12T^{2} \)
61 \( 1 + 1.66e6T + 3.14e12T^{2} \)
67 \( 1 - 1.69e6T + 6.06e12T^{2} \)
71 \( 1 + 3.19e6T + 9.09e12T^{2} \)
73 \( 1 + 1.83e6T + 1.10e13T^{2} \)
79 \( 1 - 4.85e6T + 1.92e13T^{2} \)
83 \( 1 + 3.84e6T + 2.71e13T^{2} \)
89 \( 1 + 2.37e6T + 4.42e13T^{2} \)
97 \( 1 + 4.74e6T + 8.07e13T^{2} \)
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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

−11.64769916107716751134888488552, −11.21319982772869122339618769685, −9.473291860409318164752372736481, −8.834539130154347333895689680784, −7.934065153265051592763615413458, −6.86116249363622835425109262823, −4.70330233272987203415396040068, −3.44173239824496406507770334297, −1.43044142206764317448831476794, 0, 1.43044142206764317448831476794, 3.44173239824496406507770334297, 4.70330233272987203415396040068, 6.86116249363622835425109262823, 7.934065153265051592763615413458, 8.834539130154347333895689680784, 9.473291860409318164752372736481, 11.21319982772869122339618769685, 11.64769916107716751134888488552

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