Properties

Label 2-91-1.1-c7-0-25
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 + 14.0·3-s + 5.35·4-s + 200.·5-s − 162.·6-s − 343·7-s + 1.41e3·8-s − 1.98e3·9-s − 2.31e3·10-s − 588.·11-s + 75.3·12-s + 2.19e3·13-s + 3.96e3·14-s + 2.82e3·15-s − 1.70e4·16-s + 2.00e4·17-s + 2.29e4·18-s − 9.98e3·19-s + 1.07e3·20-s − 4.82e3·21-s + 6.79e3·22-s + 4.22e4·23-s + 1.99e4·24-s − 3.78e4·25-s − 2.53e4·26-s − 5.87e4·27-s − 1.83e3·28-s + ⋯
L(s)  = 1  − 1.02·2-s + 0.300·3-s + 0.0418·4-s + 0.718·5-s − 0.307·6-s − 0.377·7-s + 0.978·8-s − 0.909·9-s − 0.733·10-s − 0.133·11-s + 0.0125·12-s + 0.277·13-s + 0.385·14-s + 0.216·15-s − 1.04·16-s + 0.987·17-s + 0.928·18-s − 0.333·19-s + 0.0300·20-s − 0.113·21-s + 0.136·22-s + 0.724·23-s + 0.294·24-s − 0.483·25-s − 0.283·26-s − 0.574·27-s − 0.0158·28-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: Trivial
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 \( 1 - 14.0T + 2.18e3T^{2} \)
5 \( 1 - 200.T + 7.81e4T^{2} \)
11 \( 1 + 588.T + 1.94e7T^{2} \)
17 \( 1 - 2.00e4T + 4.10e8T^{2} \)
19 \( 1 + 9.98e3T + 8.93e8T^{2} \)
23 \( 1 - 4.22e4T + 3.40e9T^{2} \)
29 \( 1 - 1.63e5T + 1.72e10T^{2} \)
31 \( 1 - 1.26e5T + 2.75e10T^{2} \)
37 \( 1 + 4.80e5T + 9.49e10T^{2} \)
41 \( 1 + 6.00e5T + 1.94e11T^{2} \)
43 \( 1 + 9.80e5T + 2.71e11T^{2} \)
47 \( 1 + 1.79e5T + 5.06e11T^{2} \)
53 \( 1 + 6.63e5T + 1.17e12T^{2} \)
59 \( 1 + 1.90e6T + 2.48e12T^{2} \)
61 \( 1 + 2.41e6T + 3.14e12T^{2} \)
67 \( 1 + 3.97e6T + 6.06e12T^{2} \)
71 \( 1 - 3.28e6T + 9.09e12T^{2} \)
73 \( 1 - 6.36e5T + 1.10e13T^{2} \)
79 \( 1 - 2.10e6T + 1.92e13T^{2} \)
83 \( 1 - 2.27e6T + 2.71e13T^{2} \)
89 \( 1 + 1.81e6T + 4.42e13T^{2} \)
97 \( 1 + 1.02e7T + 8.07e13T^{2} \)
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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

−12.03258981286826559777608033114, −10.59958441303215504960769220819, −9.819805660775348549509346203680, −8.828201808625923530539045632174, −8.020404989357962648987939144792, −6.50508599421838992184709863061, −5.10347929398800207091980518823, −3.15703091552755063106793971437, −1.54950365524627196511265655647, 0, 1.54950365524627196511265655647, 3.15703091552755063106793971437, 5.10347929398800207091980518823, 6.50508599421838992184709863061, 8.020404989357962648987939144792, 8.828201808625923530539045632174, 9.819805660775348549509346203680, 10.59958441303215504960769220819, 12.03258981286826559777608033114

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