Properties

Label 2-91-91.4-c1-0-1
Degree $2$
Conductor $91$
Sign $0.834 - 0.551i$
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  − 0.180i·2-s + (0.913 + 1.58i)3-s + 1.96·4-s + (−2.32 + 1.34i)5-s + (0.285 − 0.165i)6-s + (−2.46 − 0.967i)7-s − 0.717i·8-s + (−0.167 + 0.289i)9-s + (0.242 + 0.420i)10-s + (2.33 − 1.34i)11-s + (1.79 + 3.11i)12-s + (1.92 − 3.05i)13-s + (−0.174 + 0.445i)14-s + (−4.24 − 2.45i)15-s + 3.80·16-s − 4.76·17-s + ⋯
L(s)  = 1  − 0.127i·2-s + (0.527 + 0.913i)3-s + 0.983·4-s + (−1.04 + 0.600i)5-s + (0.116 − 0.0673i)6-s + (−0.930 − 0.365i)7-s − 0.253i·8-s + (−0.0557 + 0.0965i)9-s + (0.0768 + 0.133i)10-s + (0.703 − 0.406i)11-s + (0.518 + 0.898i)12-s + (0.532 − 0.846i)13-s + (−0.0467 + 0.119i)14-s + (−1.09 − 0.633i)15-s + 0.951·16-s − 1.15·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.834 - 0.551i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.834 - 0.551i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08657 + 0.326583i\)
\(L(\frac12)\) \(\approx\) \(1.08657 + 0.326583i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.46 + 0.967i)T \)
13 \( 1 + (-1.92 + 3.05i)T \)
good2 \( 1 + 0.180iT - 2T^{2} \)
3 \( 1 + (-0.913 - 1.58i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.32 - 1.34i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.33 + 1.34i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 4.76T + 17T^{2} \)
19 \( 1 + (-0.163 - 0.0942i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.39T + 23T^{2} \)
29 \( 1 + (3.54 - 6.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.20 + 1.84i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.95iT - 37T^{2} \)
41 \( 1 + (-4.70 - 2.71i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.00 + 6.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.60 - 0.924i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.53 + 6.12i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 7.58iT - 59T^{2} \)
61 \( 1 + (-0.205 + 0.356i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.87 - 5.70i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.89 + 1.67i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-12.3 - 7.10i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.55 + 7.89i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16.5iT - 83T^{2} \)
89 \( 1 + 5.89iT - 89T^{2} \)
97 \( 1 + (-0.390 + 0.225i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.58768272957122665469205271387, −13.09505747844425887261813534255, −11.80567285566522508282262580131, −10.88499545421245622198503154088, −10.06895726017323947839011095502, −8.726738667998116797728923067912, −7.31544515331466757826428984552, −6.29794709352048733301797456550, −3.86502760398039012710136254043, −3.20470804234942167004027587377, 2.10531560573797820735164018391, 3.97848288246132326619559631352, 6.28199847173491619669057323411, 7.16024655789749945043547252587, 8.186117707355295235776713678945, 9.320962546863637723493971751891, 11.11804390652511524314278579675, 12.07842933616434574852906421927, 12.73310595999314207728334165161, 13.90518716341126251498465173057

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