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Properties

Label 2-91-91.76-c1-0-3
Degree $2$
Conductor $91$
Sign $0.927 + 0.372i$
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  + (0.218 − 0.813i)2-s + (−1.24 + 0.716i)3-s + (1.11 + 0.645i)4-s + (1.01 − 1.01i)5-s + (0.312 + 1.16i)6-s + (2.43 − 1.02i)7-s + (1.95 − 1.95i)8-s + (−0.472 + 0.819i)9-s + (−0.603 − 1.04i)10-s + (−4.68 − 1.25i)11-s − 1.84·12-s + (−1.04 + 3.44i)13-s + (−0.303 − 2.20i)14-s + (−0.531 + 1.98i)15-s + (0.122 + 0.212i)16-s + (−1.49 + 2.58i)17-s + ⋯
L(s)  = 1  + (0.154 − 0.575i)2-s + (−0.716 + 0.413i)3-s + (0.558 + 0.322i)4-s + (0.453 − 0.453i)5-s + (0.127 + 0.476i)6-s + (0.921 − 0.388i)7-s + (0.692 − 0.692i)8-s + (−0.157 + 0.273i)9-s + (−0.190 − 0.330i)10-s + (−1.41 − 0.378i)11-s − 0.533·12-s + (−0.291 + 0.956i)13-s + (−0.0811 − 0.590i)14-s + (−0.137 + 0.512i)15-s + (0.0306 + 0.0531i)16-s + (−0.361 + 0.626i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.927 + 0.372i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.927 + 0.372i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03290 - 0.199691i\)
\(L(\frac12)\) \(\approx\) \(1.03290 - 0.199691i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.43 + 1.02i)T \)
13 \( 1 + (1.04 - 3.44i)T \)
good2 \( 1 + (-0.218 + 0.813i)T + (-1.73 - i)T^{2} \)
3 \( 1 + (1.24 - 0.716i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.01 + 1.01i)T - 5iT^{2} \)
11 \( 1 + (4.68 + 1.25i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.49 - 2.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.62 + 6.06i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.02 - 0.590i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.77 + 4.81i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.79 - 4.79i)T - 31iT^{2} \)
37 \( 1 + (-5.40 - 1.44i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (4.71 + 1.26i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.90 + 1.67i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.12 - 4.12i)T + 47iT^{2} \)
53 \( 1 - 5.79T + 53T^{2} \)
59 \( 1 + (-10.7 + 2.88i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-7.95 - 4.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.508 - 1.89i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.19 - 0.855i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.353 - 0.353i)T + 73iT^{2} \)
79 \( 1 + 6.95T + 79T^{2} \)
83 \( 1 + (3.22 - 3.22i)T - 83iT^{2} \)
89 \( 1 + (-0.0636 + 0.237i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.43 + 9.08i)T + (-84.0 + 48.5i)T^{2} \)
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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

−13.65211434158465534544663499952, −12.95986311807488377064779741415, −11.51052392330103968641390937290, −11.05958472052096884170383951235, −10.14546138080649793418234982491, −8.468135391084492660503096559746, −7.17916011003458386151542202349, −5.48609023512459293403130543435, −4.40697062306992403344645205162, −2.20852673932104709761642285869, 2.28621558111997335157873377981, 5.25612324869529437969997395281, 5.85756942106038820207181770810, 7.18050552087846032322130261585, 8.147331005758559507434560410435, 10.15694152137106116828959511140, 10.94935305987438357295947841797, 11.97217324207169146448268721528, 13.09554497179421092910963129132, 14.55243160433627207147803887295

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