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Properties

Label 2-91-91.6-c1-0-6
Degree $2$
Conductor $91$
Sign $-0.651 + 0.758i$
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  + (−0.357 − 1.33i)2-s + (−0.928 − 0.536i)3-s + (0.0814 − 0.0470i)4-s + (−2.02 − 2.02i)5-s + (−0.383 + 1.42i)6-s + (1.39 + 2.25i)7-s + (−2.04 − 2.04i)8-s + (−0.925 − 1.60i)9-s + (−1.98 + 3.43i)10-s + (1.37 − 0.369i)11-s − 0.100·12-s + (3.54 + 0.634i)13-s + (2.50 − 2.65i)14-s + (0.796 + 2.97i)15-s + (−1.90 + 3.29i)16-s + (2.09 + 3.63i)17-s + ⋯
L(s)  = 1  + (−0.252 − 0.942i)2-s + (−0.536 − 0.309i)3-s + (0.0407 − 0.0235i)4-s + (−0.907 − 0.907i)5-s + (−0.156 + 0.583i)6-s + (0.525 + 0.850i)7-s + (−0.722 − 0.722i)8-s + (−0.308 − 0.534i)9-s + (−0.626 + 1.08i)10-s + (0.415 − 0.111i)11-s − 0.0291·12-s + (0.984 + 0.176i)13-s + (0.669 − 0.710i)14-s + (0.205 + 0.767i)15-s + (−0.475 + 0.823i)16-s + (0.509 + 0.881i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.651 + 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.651 + 0.758i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.651 + 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.305294 - 0.664267i\)
\(L(\frac12)\) \(\approx\) \(0.305294 - 0.664267i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.39 - 2.25i)T \)
13 \( 1 + (-3.54 - 0.634i)T \)
good2 \( 1 + (0.357 + 1.33i)T + (-1.73 + i)T^{2} \)
3 \( 1 + (0.928 + 0.536i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.02 + 2.02i)T + 5iT^{2} \)
11 \( 1 + (-1.37 + 0.369i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.09 - 3.63i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.59 + 5.95i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-6.77 - 3.91i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.441 + 0.764i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.648 + 0.648i)T + 31iT^{2} \)
37 \( 1 + (7.19 - 1.92i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (11.4 - 3.07i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.809 + 0.467i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.20 + 2.20i)T - 47iT^{2} \)
53 \( 1 - 2.52T + 53T^{2} \)
59 \( 1 + (-5.65 - 1.51i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.0739 - 0.0427i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.266 - 0.995i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.79 - 0.750i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.01 + 2.01i)T - 73iT^{2} \)
79 \( 1 - 9.43T + 79T^{2} \)
83 \( 1 + (-1.54 - 1.54i)T + 83iT^{2} \)
89 \( 1 + (-1.27 - 4.75i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.37 - 8.87i)T + (-84.0 - 48.5i)T^{2} \)
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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

−13.19677188028463286753119580555, −12.13771329743694368623628846424, −11.69034202065876372315702016043, −10.96040870516971318636498346733, −9.185966005036770188793856509952, −8.522811155370048915824276114108, −6.69809712104382710190777754687, −5.32274336653629625975925089007, −3.48910084933742108070010356508, −1.19055669561669543285080200445, 3.48283892517808181016051714743, 5.24477308351069429344522097799, 6.71093401928164843758053032566, 7.54945991365061242182852210365, 8.491494811085235824092013182467, 10.45352433507534109621292031727, 11.18869266528496864446542231518, 11.97362908480266969352391259777, 13.92294440902716535636042962292, 14.65138643321781747142460169522

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