Properties

Label 2-91-13.10-c7-0-1
Degree $2$
Conductor $91$
Sign $-0.244 - 0.969i$
Analytic cond. $28.4270$
Root an. cond. $5.33170$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.87 + 3.39i)2-s + (−26.5 − 46.0i)3-s + (−40.9 + 70.9i)4-s − 36.3i·5-s + (312. + 180. i)6-s + (−297. − 171.5i)7-s − 1.42e3i·8-s + (−317. + 549. i)9-s + (123. + 213. i)10-s + (−1.69e3 + 979. i)11-s + 4.35e3·12-s + (−2.03e3 − 7.65e3i)13-s + 2.32e3·14-s + (−1.67e3 + 965. i)15-s + (−407. − 704. i)16-s + (1.02e4 − 1.78e4i)17-s + ⋯
L(s)  = 1  + (−0.519 + 0.299i)2-s + (−0.567 − 0.983i)3-s + (−0.320 + 0.554i)4-s − 0.130i·5-s + (0.590 + 0.340i)6-s + (−0.327 − 0.188i)7-s − 0.983i·8-s + (−0.145 + 0.251i)9-s + (0.0390 + 0.0675i)10-s + (−0.384 + 0.221i)11-s + 0.726·12-s + (−0.256 − 0.966i)13-s + 0.226·14-s + (−0.127 + 0.0738i)15-s + (−0.0248 − 0.0430i)16-s + (0.507 − 0.879i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.244 - 0.969i$
Analytic conductor: \(28.4270\)
Root analytic conductor: \(5.33170\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :7/2),\ -0.244 - 0.969i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.0871716 + 0.111839i\)
\(L(\frac12)\) \(\approx\) \(0.0871716 + 0.111839i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (297. + 171.5i)T \)
13 \( 1 + (2.03e3 + 7.65e3i)T \)
good2 \( 1 + (5.87 - 3.39i)T + (64 - 110. i)T^{2} \)
3 \( 1 + (26.5 + 46.0i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + 36.3iT - 7.81e4T^{2} \)
11 \( 1 + (1.69e3 - 979. i)T + (9.74e6 - 1.68e7i)T^{2} \)
17 \( 1 + (-1.02e4 + 1.78e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (3.63e4 + 2.10e4i)T + (4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (9.80e3 + 1.69e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + (-3.89e4 - 6.73e4i)T + (-8.62e9 + 1.49e10i)T^{2} \)
31 \( 1 - 1.66e5iT - 2.75e10T^{2} \)
37 \( 1 + (-6.10e4 + 3.52e4i)T + (4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 + (4.63e5 - 2.67e5i)T + (9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + (5.62e4 - 9.74e4i)T + (-1.35e11 - 2.35e11i)T^{2} \)
47 \( 1 - 1.21e6iT - 5.06e11T^{2} \)
53 \( 1 - 2.06e5T + 1.17e12T^{2} \)
59 \( 1 + (2.42e6 + 1.40e6i)T + (1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (1.42e6 - 2.47e6i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (-2.11e6 + 1.22e6i)T + (3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + (-3.43e6 - 1.98e6i)T + (4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 + 3.59e6iT - 1.10e13T^{2} \)
79 \( 1 - 3.66e6T + 1.92e13T^{2} \)
83 \( 1 - 8.26e6iT - 2.71e13T^{2} \)
89 \( 1 + (-4.68e6 + 2.70e6i)T + (2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + (8.60e6 + 4.96e6i)T + (4.03e13 + 6.99e13i)T^{2} \)
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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

−12.61975286887518401052335472015, −12.43733019027981811610047928601, −10.80615642221869319397811322671, −9.554670167635654788139137984228, −8.306300178420424035092720823907, −7.30833342558818995422304271946, −6.48616245387278209272584568330, −4.83858572429033739672796738154, −2.99798290137362052108552417140, −0.919550299917202933981521298251, 0.07666978419663924286565706184, 1.96464991146772718200157766752, 4.01325519650485451084448452925, 5.19893008614984146715093079707, 6.28915084804527125457131419246, 8.236659226433378527879207652835, 9.391771472863721127031375107243, 10.27200339729202639322402427338, 10.85159675835110552811539189001, 12.03318836387975348136798438599

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