Properties

Label 2-91-13.9-c7-0-39
Degree $2$
Conductor $91$
Sign $-0.645 + 0.763i$
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  + (8.46 − 14.6i)2-s + (17.8 − 30.9i)3-s + (−79.4 − 137. i)4-s + 457.·5-s + (−302. − 524. i)6-s + (171.5 + 297. i)7-s − 522.·8-s + (454. + 786. i)9-s + (3.87e3 − 6.71e3i)10-s + (559. − 969. i)11-s − 5.68e3·12-s + (3.32e3 − 7.18e3i)13-s + 5.80e3·14-s + (8.17e3 − 1.41e4i)15-s + (5.73e3 − 9.94e3i)16-s + (4.75e3 + 8.24e3i)17-s + ⋯
L(s)  = 1  + (0.748 − 1.29i)2-s + (0.382 − 0.662i)3-s + (−0.620 − 1.07i)4-s + 1.63·5-s + (−0.572 − 0.991i)6-s + (0.188 + 0.327i)7-s − 0.361·8-s + (0.207 + 0.359i)9-s + (1.22 − 2.12i)10-s + (0.126 − 0.219i)11-s − 0.948·12-s + (0.420 − 0.907i)13-s + 0.565·14-s + (0.625 − 1.08i)15-s + (0.350 − 0.606i)16-s + (0.234 + 0.406i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.645 + 0.763i$
Analytic conductor: \(28.4270\)
Root analytic conductor: \(5.33170\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :7/2),\ -0.645 + 0.763i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.02713 - 4.36808i\)
\(L(\frac12)\) \(\approx\) \(2.02713 - 4.36808i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-171.5 - 297. i)T \)
13 \( 1 + (-3.32e3 + 7.18e3i)T \)
good2 \( 1 + (-8.46 + 14.6i)T + (-64 - 110. i)T^{2} \)
3 \( 1 + (-17.8 + 30.9i)T + (-1.09e3 - 1.89e3i)T^{2} \)
5 \( 1 - 457.T + 7.81e4T^{2} \)
11 \( 1 + (-559. + 969. i)T + (-9.74e6 - 1.68e7i)T^{2} \)
17 \( 1 + (-4.75e3 - 8.24e3i)T + (-2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (8.06e3 + 1.39e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (4.38e4 - 7.60e4i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 + (-5.58e3 + 9.67e3i)T + (-8.62e9 - 1.49e10i)T^{2} \)
31 \( 1 + 2.91e5T + 2.75e10T^{2} \)
37 \( 1 + (-9.17e4 + 1.58e5i)T + (-4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 + (3.21e5 - 5.57e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + (-2.72e5 - 4.71e5i)T + (-1.35e11 + 2.35e11i)T^{2} \)
47 \( 1 + 1.22e6T + 5.06e11T^{2} \)
53 \( 1 + 8.49e5T + 1.17e12T^{2} \)
59 \( 1 + (-1.50e6 - 2.60e6i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (1.76e6 + 3.06e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (3.17e5 - 5.50e5i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + (1.64e6 + 2.85e6i)T + (-4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 - 3.29e6T + 1.10e13T^{2} \)
79 \( 1 + 5.67e5T + 1.92e13T^{2} \)
83 \( 1 - 5.08e6T + 2.71e13T^{2} \)
89 \( 1 + (-2.71e6 + 4.70e6i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + (-2.60e6 - 4.50e6i)T + (-4.03e13 + 6.99e13i)T^{2} \)
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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

−12.72244775609821630214284228493, −11.25962818099701690021044024518, −10.32535183988840881856064668089, −9.363264625716165483305520082577, −7.83739071988535454920720865144, −6.07393890500774785260655504184, −5.06682653728595495946901885531, −3.21077810133642138776277039197, −2.02937939334443700673485214692, −1.39660366364747744871421019126, 1.79699264273918409910249780886, 3.83554752531985777535597243337, 4.96850631024781740130857911349, 6.13516251691608848712738347965, 6.94132154394453577796686293599, 8.608963066392313768323838534710, 9.626103139461429567512672405202, 10.56735694321476439203875885205, 12.52238745693670661578059803337, 13.58656658893372669120362747199

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