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Properties

Label 2-91-91.20-c1-0-2
Degree $2$
Conductor $91$
Sign $0.602 - 0.798i$
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  + (−2.47 + 0.664i)2-s + (2.33 + 1.34i)3-s + (3.97 − 2.29i)4-s + (0.281 − 0.281i)5-s + (−6.68 − 1.79i)6-s + (1.14 − 2.38i)7-s + (−4.70 + 4.70i)8-s + (2.13 + 3.69i)9-s + (−0.511 + 0.885i)10-s + (0.939 + 3.50i)11-s + 12.3·12-s + (−3.44 + 1.06i)13-s + (−1.25 + 6.67i)14-s + (1.03 − 0.277i)15-s + (3.95 − 6.84i)16-s + (−2.04 − 3.54i)17-s + ⋯
L(s)  = 1  + (−1.75 + 0.469i)2-s + (1.34 + 0.778i)3-s + (1.98 − 1.14i)4-s + (0.125 − 0.125i)5-s + (−2.72 − 0.731i)6-s + (0.432 − 0.901i)7-s + (−1.66 + 1.66i)8-s + (0.711 + 1.23i)9-s + (−0.161 + 0.280i)10-s + (0.283 + 1.05i)11-s + 3.57·12-s + (−0.955 + 0.294i)13-s + (−0.334 + 1.78i)14-s + (0.267 − 0.0717i)15-s + (0.987 − 1.71i)16-s + (−0.496 − 0.860i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 - 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.602 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.602 - 0.798i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.602 - 0.798i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.635075 + 0.316475i\)
\(L(\frac12)\) \(\approx\) \(0.635075 + 0.316475i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.14 + 2.38i)T \)
13 \( 1 + (3.44 - 1.06i)T \)
good2 \( 1 + (2.47 - 0.664i)T + (1.73 - i)T^{2} \)
3 \( 1 + (-2.33 - 1.34i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.281 + 0.281i)T - 5iT^{2} \)
11 \( 1 + (-0.939 - 3.50i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.04 + 3.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.777 + 0.208i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.41 + 2.54i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.00 + 1.74i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.44 + 4.44i)T - 31iT^{2} \)
37 \( 1 + (-0.463 - 1.73i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.578 + 2.15i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.65 - 1.53i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.99 + 5.99i)T + 47iT^{2} \)
53 \( 1 - 9.09T + 53T^{2} \)
59 \( 1 + (1.92 - 7.17i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.40 - 1.38i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.85 - 1.30i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.582 - 2.17i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-3.50 - 3.50i)T + 73iT^{2} \)
79 \( 1 + 3.61T + 79T^{2} \)
83 \( 1 + (5.36 - 5.36i)T - 83iT^{2} \)
89 \( 1 + (-11.5 + 3.09i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-7.71 - 2.06i)T + (84.0 + 48.5i)T^{2} \)
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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

−14.75124352045353202376742547815, −13.65371990230142130552354868233, −11.63497023421146438359647123720, −10.17008217429410668691160185942, −9.780182303479263769936214601243, −8.821645923429909444118612903169, −7.78615451305835779249856947290, −6.97896827767333305838631450897, −4.47904762946619972745953370140, −2.19783988813740389148499081828, 1.87384956609549070413691502941, 2.96902637538223621263211813221, 6.45350917498622160528797917725, 7.87116018662574451790819557380, 8.413686603938450181813788312248, 9.192067809338529194924694690267, 10.36743577148396596540362460707, 11.69308488057990225490873423729, 12.58726446861009141254309150056, 14.01212406249623897319392468088

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