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Properties

Label 2-91-13.10-c1-0-0
Degree $2$
Conductor $91$
Sign $-0.383 - 0.923i$
Analytic cond. $0.726638$
Root an. cond. $0.852431$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.20 + 0.692i)2-s + (1.41 + 2.44i)3-s + (−0.0395 + 0.0685i)4-s − 0.518i·5-s + (−3.39 − 1.95i)6-s + (−0.866 − 0.5i)7-s − 2.88i·8-s + (−2.49 + 4.31i)9-s + (0.359 + 0.622i)10-s + (1.40 − 0.812i)11-s − 0.223·12-s + (1.42 + 3.31i)13-s + 1.38·14-s + (1.26 − 0.733i)15-s + (1.91 + 3.32i)16-s + (0.974 − 1.68i)17-s + ⋯
L(s)  = 1  + (−0.848 + 0.490i)2-s + (0.815 + 1.41i)3-s + (−0.0197 + 0.0342i)4-s − 0.232i·5-s + (−1.38 − 0.799i)6-s + (−0.327 − 0.188i)7-s − 1.01i·8-s + (−0.830 + 1.43i)9-s + (0.113 + 0.196i)10-s + (0.424 − 0.244i)11-s − 0.0645·12-s + (0.395 + 0.918i)13-s + 0.370·14-s + (0.327 − 0.189i)15-s + (0.479 + 0.830i)16-s + (0.236 − 0.409i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.383 - 0.923i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.383 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.432642 + 0.648347i\)
\(L(\frac12)\) \(\approx\) \(0.432642 + 0.648347i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-1.42 - 3.31i)T \)
good2 \( 1 + (1.20 - 0.692i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.41 - 2.44i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 0.518iT - 5T^{2} \)
11 \( 1 + (-1.40 + 0.812i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.974 + 1.68i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.15 - 1.24i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.57 + 7.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.61 - 4.52i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.79iT - 31T^{2} \)
37 \( 1 + (8.85 - 5.11i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.64 + 2.10i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.498 - 0.863i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.51iT - 47T^{2} \)
53 \( 1 + 8.89T + 53T^{2} \)
59 \( 1 + (5.37 + 3.10i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.73 + 11.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.25 - 4.18i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.50 + 2.59i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 11.8iT - 73T^{2} \)
79 \( 1 - 0.982T + 79T^{2} \)
83 \( 1 - 8.91iT - 83T^{2} \)
89 \( 1 + (10.4 - 6.00i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.82 - 2.21i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.56318564934029895581166298795, −13.77417865246612545996832831438, −12.27393795288480786418426784612, −10.64757254357678628578872970457, −9.701002453378307126912968250179, −8.962296011531587274047733800958, −8.203943044969050086982157961538, −6.63589867357642471318704246595, −4.54816144972298883613147924171, −3.43145108164336054455557537863, 1.45390583995939883720046521229, 3.02083321004080896155170308069, 5.83677852176420868914421980283, 7.28173288126409391739064551400, 8.260960236549061719177728715997, 9.188479679466817661547777377150, 10.34315023876456713962016632621, 11.69950957571609746995082346777, 12.67448349900212106314621502601, 13.75452316152219879934118989729

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