Properties

Label 2-91-1.1-c9-0-24
Degree $2$
Conductor $91$
Sign $-1$
Analytic cond. $46.8682$
Root an. cond. $6.84603$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 39.6·2-s − 201.·3-s + 1.06e3·4-s + 764.·5-s + 8.00e3·6-s + 2.40e3·7-s − 2.18e4·8-s + 2.10e4·9-s − 3.03e4·10-s − 8.70e4·11-s − 2.14e5·12-s + 2.85e4·13-s − 9.52e4·14-s − 1.54e5·15-s + 3.22e5·16-s − 4.61e4·17-s − 8.35e5·18-s + 5.04e5·19-s + 8.12e5·20-s − 4.84e5·21-s + 3.45e6·22-s − 5.99e5·23-s + 4.40e6·24-s − 1.36e6·25-s − 1.13e6·26-s − 2.76e5·27-s + 2.55e6·28-s + ⋯
L(s)  = 1  − 1.75·2-s − 1.43·3-s + 2.07·4-s + 0.546·5-s + 2.52·6-s + 0.377·7-s − 1.88·8-s + 1.06·9-s − 0.959·10-s − 1.79·11-s − 2.98·12-s + 0.277·13-s − 0.662·14-s − 0.786·15-s + 1.23·16-s − 0.134·17-s − 1.87·18-s + 0.888·19-s + 1.13·20-s − 0.543·21-s + 3.14·22-s − 0.446·23-s + 2.71·24-s − 0.700·25-s − 0.486·26-s − 0.100·27-s + 0.784·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(46.8682\)
Root analytic conductor: \(6.84603\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 91,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - 2.40e3T \)
13 \( 1 - 2.85e4T \)
good2 \( 1 + 39.6T + 512T^{2} \)
3 \( 1 + 201.T + 1.96e4T^{2} \)
5 \( 1 - 764.T + 1.95e6T^{2} \)
11 \( 1 + 8.70e4T + 2.35e9T^{2} \)
17 \( 1 + 4.61e4T + 1.18e11T^{2} \)
19 \( 1 - 5.04e5T + 3.22e11T^{2} \)
23 \( 1 + 5.99e5T + 1.80e12T^{2} \)
29 \( 1 - 2.58e6T + 1.45e13T^{2} \)
31 \( 1 - 3.96e6T + 2.64e13T^{2} \)
37 \( 1 + 6.19e6T + 1.29e14T^{2} \)
41 \( 1 + 1.66e7T + 3.27e14T^{2} \)
43 \( 1 - 3.31e7T + 5.02e14T^{2} \)
47 \( 1 - 2.64e7T + 1.11e15T^{2} \)
53 \( 1 + 3.90e7T + 3.29e15T^{2} \)
59 \( 1 - 4.81e7T + 8.66e15T^{2} \)
61 \( 1 - 1.43e8T + 1.16e16T^{2} \)
67 \( 1 + 1.77e7T + 2.72e16T^{2} \)
71 \( 1 + 8.31e7T + 4.58e16T^{2} \)
73 \( 1 - 2.29e8T + 5.88e16T^{2} \)
79 \( 1 + 2.99e8T + 1.19e17T^{2} \)
83 \( 1 - 8.34e8T + 1.86e17T^{2} \)
89 \( 1 + 2.13e8T + 3.50e17T^{2} \)
97 \( 1 + 1.36e9T + 7.60e17T^{2} \)
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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

−11.25802365071237522677960872965, −10.48288403767470548980079761797, −9.852023700021116824801156792254, −8.367446488467166015682593224959, −7.37061794508092070714796371796, −6.11604079431850129466228299444, −5.14335896887931214894494143410, −2.38465508933771413308684296504, −1.01677652399347528943642336579, 0, 1.01677652399347528943642336579, 2.38465508933771413308684296504, 5.14335896887931214894494143410, 6.11604079431850129466228299444, 7.37061794508092070714796371796, 8.367446488467166015682593224959, 9.852023700021116824801156792254, 10.48288403767470548980079761797, 11.25802365071237522677960872965

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