Properties

Label 2-91-91.19-c1-0-6
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: $\chi_{91} (19, \cdot )$
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

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

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