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Properties

Label 2-91-91.24-c1-0-0
Degree $2$
Conductor $91$
Sign $0.0845 - 0.996i$
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  + (0.263 − 0.0707i)2-s + 2.50i·3-s + (−1.66 + 0.962i)4-s + (−1.43 − 0.383i)5-s + (0.177 + 0.661i)6-s + (2.63 + 0.252i)7-s + (−0.758 + 0.758i)8-s − 3.28·9-s − 0.404·10-s + (3.24 − 3.24i)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 + (1.77 − 3.08i)16-s + (1.17 + 2.02i)17-s + ⋯
L(s)  = 1  + (0.186 − 0.0500i)2-s + 1.44i·3-s + (−0.833 + 0.481i)4-s + (−0.639 − 0.171i)5-s + (0.0723 + 0.270i)6-s + (0.995 + 0.0953i)7-s + (−0.268 + 0.268i)8-s − 1.09·9-s − 0.127·10-s + (0.978 − 0.978i)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 + (0.444 − 0.770i)16-s + (0.283 + 0.491i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0845 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0845 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.0845 - 0.996i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.0845 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.681891 + 0.626479i\)
\(L(\frac12)\) \(\approx\) \(0.681891 + 0.626479i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.63 - 0.252i)T \)
13 \( 1 + (-2.80 - 2.26i)T \)
good2 \( 1 + (-0.263 + 0.0707i)T + (1.73 - i)T^{2} \)
3 \( 1 - 2.50iT - 3T^{2} \)
5 \( 1 + (1.43 + 0.383i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-3.24 + 3.24i)T - 11iT^{2} \)
17 \( 1 + (-1.17 - 2.02i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.19 - 1.19i)T - 19iT^{2} \)
23 \( 1 + (4.15 + 2.39i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.87 + 4.98i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.52 + 5.69i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.20 - 8.24i)T + (-32.0 + 18.5i)T^{2} \)
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 \( 1 + (-2.07 + 7.75i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.54 + 11.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.400 - 1.49i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 - 9.76iT - 61T^{2} \)
67 \( 1 + (0.385 + 0.385i)T + 67iT^{2} \)
71 \( 1 + (-1.77 + 0.474i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.28 - 0.611i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.13 + 3.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.88 - 3.88i)T - 83iT^{2} \)
89 \( 1 + (13.5 - 3.63i)T + (77.0 - 44.5i)T^{2} \)
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

−14.45784765198306482419141628476, −13.54999107709969104014362066296, −11.86271214299803089073660156025, −11.30917429138666953135864475530, −9.916081394464450327837767584029, −8.732867457854926437850345380554, −8.177933897949261265349903452586, −5.75192076198319891466946933276, −4.20448036530025056717980853092, −3.90484919304841728972145218346, 1.42655261271394217811437632498, 4.09113981992697472688574533733, 5.66931523627900458322609473586, 7.11391723800311029942471716453, 7.998563330866759198111918566992, 9.197884722331548552621521122046, 10.85923845357453877049127408335, 12.00757446638207646249517054338, 12.77667772261079680504276033703, 13.91844039732085367406861650312

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