Properties

Label 2-91-1.1-c9-0-28
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  − 37.8·2-s + 32.9·3-s + 918.·4-s − 2.42e3·5-s − 1.24e3·6-s + 2.40e3·7-s − 1.53e4·8-s − 1.85e4·9-s + 9.18e4·10-s + 3.85e4·11-s + 3.03e4·12-s + 2.85e4·13-s − 9.08e4·14-s − 8.01e4·15-s + 1.11e5·16-s − 1.87e5·17-s + 7.03e5·18-s − 1.09e5·19-s − 2.23e6·20-s + 7.92e4·21-s − 1.45e6·22-s + 5.99e5·23-s − 5.07e5·24-s + 3.94e6·25-s − 1.08e6·26-s − 1.26e6·27-s + 2.20e6·28-s + ⋯
L(s)  = 1  − 1.67·2-s + 0.235·3-s + 1.79·4-s − 1.73·5-s − 0.393·6-s + 0.377·7-s − 1.32·8-s − 0.944·9-s + 2.90·10-s + 0.793·11-s + 0.422·12-s + 0.277·13-s − 0.631·14-s − 0.408·15-s + 0.426·16-s − 0.543·17-s + 1.57·18-s − 0.193·19-s − 3.11·20-s + 0.0889·21-s − 1.32·22-s + 0.446·23-s − 0.312·24-s + 2.01·25-s − 0.463·26-s − 0.457·27-s + 0.678·28-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 \( 1 + 37.8T + 512T^{2} \)
3 \( 1 - 32.9T + 1.96e4T^{2} \)
5 \( 1 + 2.42e3T + 1.95e6T^{2} \)
11 \( 1 - 3.85e4T + 2.35e9T^{2} \)
17 \( 1 + 1.87e5T + 1.18e11T^{2} \)
19 \( 1 + 1.09e5T + 3.22e11T^{2} \)
23 \( 1 - 5.99e5T + 1.80e12T^{2} \)
29 \( 1 - 1.91e6T + 1.45e13T^{2} \)
31 \( 1 - 9.83e6T + 2.64e13T^{2} \)
37 \( 1 + 6.01e6T + 1.29e14T^{2} \)
41 \( 1 + 2.93e6T + 3.27e14T^{2} \)
43 \( 1 - 3.42e7T + 5.02e14T^{2} \)
47 \( 1 - 4.06e7T + 1.11e15T^{2} \)
53 \( 1 + 1.10e8T + 3.29e15T^{2} \)
59 \( 1 + 3.51e7T + 8.66e15T^{2} \)
61 \( 1 + 4.03e7T + 1.16e16T^{2} \)
67 \( 1 - 9.71e7T + 2.72e16T^{2} \)
71 \( 1 + 3.17e8T + 4.58e16T^{2} \)
73 \( 1 - 1.93e6T + 5.88e16T^{2} \)
79 \( 1 - 2.55e8T + 1.19e17T^{2} \)
83 \( 1 + 6.62e8T + 1.86e17T^{2} \)
89 \( 1 + 1.50e7T + 3.50e17T^{2} \)
97 \( 1 - 7.63e8T + 7.60e17T^{2} \)
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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

−11.41939576966240107514531021440, −10.71501159251808113991337713943, −9.098765541431493970190682460017, −8.427033974589085136572503562366, −7.70571554280565585294307649162, −6.56606876086326646733388412462, −4.32260428109130752430666227245, −2.82837065619745434634106761884, −1.06301772263503479287313519516, 0, 1.06301772263503479287313519516, 2.82837065619745434634106761884, 4.32260428109130752430666227245, 6.56606876086326646733388412462, 7.70571554280565585294307649162, 8.427033974589085136572503562366, 9.098765541431493970190682460017, 10.71501159251808113991337713943, 11.41939576966240107514531021440

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