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Properties

Label 2-91-91.6-c1-0-3
Degree $2$
Conductor $91$
Sign $0.718 + 0.695i$
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 + (−2.32 + 1.25i)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 + (−0.880 + 0.473i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.718 + 0.695i$
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: \((2,\ 91,\ (\ :1/2),\ 0.718 + 0.695i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00417 - 0.406393i\)
\(L(\frac12)\) \(\approx\) \(1.00417 - 0.406393i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.32 - 1.25i)T \)
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

−14.04808417716519391588111246553, −12.69554848717986018201854692045, −11.75254016350530410260211924601, −10.51323947008924305427600854891, −9.651399022717627665430651524662, −9.122288847692069748165466664089, −6.91367343571081113995775427650, −5.92808023983707360040315939324, −3.34364895138901415043640605549, −2.46548901224625454702084300880, 2.51329131574347318885492636273, 4.96830798912295256050122777996, 6.39977368858164879944507779495, 7.32212390750980612878806198290, 8.728861340738151411239547676635, 9.288089495579267626280685446329, 10.86031049952502627893444815668, 12.56055394360776911842395156524, 13.24257101016389806392186622463, 14.28330854156036063984453134849

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