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Properties

Label 2-91-91.89-c1-0-4
Degree $2$
Conductor $91$
Sign $0.989 + 0.146i$
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  + (−0.193 − 0.193i)2-s + (2.17 + 1.25i)3-s − 1.92i·4-s + (−0.383 − 1.43i)5-s + (−0.177 − 0.661i)6-s + (−2.15 + 1.53i)7-s + (−0.758 + 0.758i)8-s + (1.64 + 2.84i)9-s + (−0.202 + 0.350i)10-s + (1.18 + 4.43i)11-s + (2.41 − 4.17i)12-s + (−2.80 − 2.26i)13-s + (0.713 + 0.119i)14-s + (0.960 − 3.58i)15-s − 3.55·16-s + 2.34·17-s + ⋯
L(s)  = 1  + (−0.136 − 0.136i)2-s + (1.25 + 0.723i)3-s − 0.962i·4-s + (−0.171 − 0.639i)5-s + (−0.0723 − 0.270i)6-s + (−0.814 + 0.580i)7-s + (−0.268 + 0.268i)8-s + (0.546 + 0.947i)9-s + (−0.0639 + 0.110i)10-s + (0.358 + 1.33i)11-s + (0.696 − 1.20i)12-s + (−0.778 − 0.627i)13-s + (0.190 + 0.0319i)14-s + (0.247 − 0.925i)15-s − 0.889·16-s + 0.567·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.989 + 0.146i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.989 + 0.146i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16650 - 0.0860107i\)
\(L(\frac12)\) \(\approx\) \(1.16650 - 0.0860107i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.15 - 1.53i)T \)
13 \( 1 + (2.80 + 2.26i)T \)
good2 \( 1 + (0.193 + 0.193i)T + 2iT^{2} \)
3 \( 1 + (-2.17 - 1.25i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.383 + 1.43i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.18 - 4.43i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 - 2.34T + 17T^{2} \)
19 \( 1 + (1.63 + 0.438i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 4.79iT - 23T^{2} \)
29 \( 1 + (2.87 + 4.98i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.69 + 1.52i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-6.03 + 6.03i)T - 37iT^{2} \)
41 \( 1 + (0.829 + 0.222i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.70 - 0.981i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.75 + 2.07i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.54 - 11.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.09 - 1.09i)T + 59iT^{2} \)
61 \( 1 + (8.45 - 4.88i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.526 + 0.141i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.77 + 0.474i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.611 - 2.28i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.13 - 3.69i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.88 + 3.88i)T - 83iT^{2} \)
89 \( 1 + (9.92 + 9.92i)T + 89iT^{2} \)
97 \( 1 + (-0.734 - 2.73i)T + (-84.0 + 48.5i)T^{2} \)
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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

−14.40787837999589717827340494051, −13.08433172733509129801652150565, −12.07686058153958198366555963939, −10.31908984875962839373317393059, −9.491343643620630587927242764637, −9.059662996977581080480347984892, −7.49685646044163981475255874468, −5.62892207371662658848844416030, −4.23515786865499447692685667200, −2.43944534125962550937920690416, 2.83413747411958218508898491720, 3.69388749206062908732665728089, 6.64650377646740955724441444755, 7.35594969013145677894647520153, 8.430127408993356405780478088563, 9.332483817883270816694353443992, 10.93612355429606537413962868718, 12.33233158047012093020450027547, 13.17160401623802486396527554914, 14.06704762483983723946358745417

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