Properties

Label 2-91-1.1-c7-0-38
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  + 19.7·2-s − 37.6·3-s + 260.·4-s − 267.·5-s − 743.·6-s + 343·7-s + 2.62e3·8-s − 766.·9-s − 5.27e3·10-s + 827.·11-s − 9.83e3·12-s − 2.19e3·13-s + 6.76e3·14-s + 1.00e4·15-s + 1.83e4·16-s − 3.35e4·17-s − 1.51e4·18-s − 4.67e4·19-s − 6.97e4·20-s − 1.29e4·21-s + 1.63e4·22-s − 1.31e4·23-s − 9.88e4·24-s − 6.68e3·25-s − 4.33e4·26-s + 1.11e5·27-s + 8.95e4·28-s + ⋯
L(s)  = 1  + 1.74·2-s − 0.806·3-s + 2.03·4-s − 0.956·5-s − 1.40·6-s + 0.377·7-s + 1.81·8-s − 0.350·9-s − 1.66·10-s + 0.187·11-s − 1.64·12-s − 0.277·13-s + 0.658·14-s + 0.770·15-s + 1.11·16-s − 1.65·17-s − 0.610·18-s − 1.56·19-s − 1.94·20-s − 0.304·21-s + 0.326·22-s − 0.224·23-s − 1.45·24-s − 0.0855·25-s − 0.483·26-s + 1.08·27-s + 0.770·28-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 \( 1 - 19.7T + 128T^{2} \)
3 \( 1 + 37.6T + 2.18e3T^{2} \)
5 \( 1 + 267.T + 7.81e4T^{2} \)
11 \( 1 - 827.T + 1.94e7T^{2} \)
17 \( 1 + 3.35e4T + 4.10e8T^{2} \)
19 \( 1 + 4.67e4T + 8.93e8T^{2} \)
23 \( 1 + 1.31e4T + 3.40e9T^{2} \)
29 \( 1 - 1.02e5T + 1.72e10T^{2} \)
31 \( 1 + 1.13e5T + 2.75e10T^{2} \)
37 \( 1 + 1.81e4T + 9.49e10T^{2} \)
41 \( 1 - 7.53e5T + 1.94e11T^{2} \)
43 \( 1 + 5.12e5T + 2.71e11T^{2} \)
47 \( 1 + 2.15e5T + 5.06e11T^{2} \)
53 \( 1 - 1.40e6T + 1.17e12T^{2} \)
59 \( 1 - 4.01e5T + 2.48e12T^{2} \)
61 \( 1 + 1.48e6T + 3.14e12T^{2} \)
67 \( 1 - 4.12e6T + 6.06e12T^{2} \)
71 \( 1 + 2.09e6T + 9.09e12T^{2} \)
73 \( 1 - 2.16e6T + 1.10e13T^{2} \)
79 \( 1 + 6.79e5T + 1.92e13T^{2} \)
83 \( 1 + 9.02e6T + 2.71e13T^{2} \)
89 \( 1 - 2.31e6T + 4.42e13T^{2} \)
97 \( 1 + 4.51e6T + 8.07e13T^{2} \)
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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

−12.18653938371114612506139278082, −11.38025396665588053080517782220, −10.82991624980821816354116779299, −8.506455862181618597373722726032, −6.96627893484159272246396201394, −6.03679296871082391098413630104, −4.76186561977956370086769896209, −4.01111773761644286317912745129, −2.36527736357680293643863215886, 0, 2.36527736357680293643863215886, 4.01111773761644286317912745129, 4.76186561977956370086769896209, 6.03679296871082391098413630104, 6.96627893484159272246396201394, 8.506455862181618597373722726032, 10.82991624980821816354116779299, 11.38025396665588053080517782220, 12.18653938371114612506139278082

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