Properties

Label 2-91-91.74-c1-0-1
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  − 0.268·2-s + (0.571 + 0.989i)3-s − 1.92·4-s + (1.28 + 2.21i)5-s + (−0.153 − 0.265i)6-s + (1.80 + 1.93i)7-s + 1.05·8-s + (0.846 − 1.46i)9-s + (−0.343 − 0.594i)10-s + (−1.97 − 3.41i)11-s + (−1.10 − 1.90i)12-s + (−3.15 + 1.74i)13-s + (−0.483 − 0.518i)14-s + (−1.46 + 2.53i)15-s + 3.57·16-s + 0.785·17-s + ⋯
L(s)  = 1  − 0.189·2-s + (0.329 + 0.571i)3-s − 0.964·4-s + (0.572 + 0.992i)5-s + (−0.0625 − 0.108i)6-s + (0.681 + 0.731i)7-s + 0.372·8-s + (0.282 − 0.488i)9-s + (−0.108 − 0.188i)10-s + (−0.594 − 1.03i)11-s + (−0.318 − 0.550i)12-s + (−0.874 + 0.484i)13-s + (−0.129 − 0.138i)14-s + (−0.378 + 0.654i)15-s + 0.893·16-s + 0.190·17-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: $\chi_{91} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.602 - 0.798i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.828708 + 0.413053i\)
\(L(\frac12)\) \(\approx\) \(0.828708 + 0.413053i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.80 - 1.93i)T \)
13 \( 1 + (3.15 - 1.74i)T \)
good2 \( 1 + 0.268T + 2T^{2} \)
3 \( 1 + (-0.571 - 0.989i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.28 - 2.21i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.97 + 3.41i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 0.785T + 17T^{2} \)
19 \( 1 + (-3.74 + 6.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.95T + 23T^{2} \)
29 \( 1 + (1.17 - 2.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.27 + 2.21i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.75T + 37T^{2} \)
41 \( 1 + (-1.21 + 2.11i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.12 - 1.94i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.658 + 1.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.63 - 8.03i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.96T + 59T^{2} \)
61 \( 1 + (4.72 - 8.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.676 - 1.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.15 + 10.6i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.384 - 0.665i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.09 + 5.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.07T + 83T^{2} \)
89 \( 1 - 7.66T + 89T^{2} \)
97 \( 1 + (-1.18 - 2.05i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.28401965928118470695504249070, −13.56809757585358019526642401355, −12.06495297127823811831190490385, −10.77234350650402327647341572402, −9.716956319428040807631666517085, −9.003898392729008372515682562890, −7.69822977920066813683498807627, −5.94441765064371907390704234184, −4.59538770372467674838971163489, −2.87269774456217944358029465116, 1.62808694153426080522566773291, 4.44274701910350009850597901794, 5.36097123867070531769312766878, 7.73161417736183810967779246323, 8.010762927816694926408780967973, 9.692418898739610595710082534028, 10.21357259771832441486613656934, 12.28495178827761896817550803731, 12.94959178670275455080497563019, 13.84389974435459931023010014142

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