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Properties

Label 2-91-91.54-c1-0-4
Degree $2$
Conductor $91$
Sign $0.918 - 0.395i$
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  + (0.203 − 0.203i)2-s + (0.923 + 0.532i)3-s + 1.91i·4-s + (0.499 − 0.133i)5-s + (0.296 − 0.0794i)6-s + (−2.60 − 0.449i)7-s + (0.798 + 0.798i)8-s + (−0.931 − 1.61i)9-s + (0.0744 − 0.129i)10-s + (3.69 − 0.990i)11-s + (−1.02 + 1.76i)12-s + (1.12 − 3.42i)13-s + (−0.622 + 0.439i)14-s + (0.532 + 0.142i)15-s − 3.50·16-s − 4.52·17-s + ⋯
L(s)  = 1  + (0.144 − 0.144i)2-s + (0.532 + 0.307i)3-s + 0.958i·4-s + (0.223 − 0.0598i)5-s + (0.121 − 0.0324i)6-s + (−0.985 − 0.169i)7-s + (0.282 + 0.282i)8-s + (−0.310 − 0.538i)9-s + (0.0235 − 0.0407i)10-s + (1.11 − 0.298i)11-s + (−0.294 + 0.510i)12-s + (0.312 − 0.950i)13-s + (−0.166 + 0.117i)14-s + (0.137 + 0.0368i)15-s − 0.877·16-s − 1.09·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.918 - 0.395i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.918 - 0.395i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14156 + 0.235387i\)
\(L(\frac12)\) \(\approx\) \(1.14156 + 0.235387i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.60 + 0.449i)T \)
13 \( 1 + (-1.12 + 3.42i)T \)
good2 \( 1 + (-0.203 + 0.203i)T - 2iT^{2} \)
3 \( 1 + (-0.923 - 0.532i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.499 + 0.133i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-3.69 + 0.990i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + 4.52T + 17T^{2} \)
19 \( 1 + (-0.797 + 2.97i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 8.67iT - 23T^{2} \)
29 \( 1 + (-1.26 - 2.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.270 + 1.00i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.129 + 0.129i)T + 37iT^{2} \)
41 \( 1 + (-1.79 + 6.68i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.59 - 2.65i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.42 - 9.03i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.512 - 0.887i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.368 + 0.368i)T - 59iT^{2} \)
61 \( 1 + (7.39 - 4.26i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.83 + 6.83i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.96 - 7.33i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (11.9 + 3.20i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.77 + 6.54i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.42 - 6.42i)T + 83iT^{2} \)
89 \( 1 + (-8.56 + 8.56i)T - 89iT^{2} \)
97 \( 1 + (13.1 - 3.53i)T + (84.0 - 48.5i)T^{2} \)
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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

−13.85222820791959294706771086819, −13.25078610583381281405747036353, −12.13216827856725074788691976320, −11.09460282990907380999085578177, −9.431590495115686760887900322132, −8.899298407630737525908274581386, −7.41871832234766103840719664920, −6.09056727341958373996318646608, −3.95021359306661374594861742803, −3.05660061842406579729998440505, 2.15767277195747218401145427645, 4.30711412896718686583683432710, 6.11317851296678870866973094152, 6.82710473650440435389575365207, 8.678503104721378123214211404581, 9.548748373263323028483321344215, 10.65759176641612815772786067387, 11.97532597705304311654030299442, 13.33000262686259053994620103681, 14.04261827133941753482021707706

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