Properties

Label 2-91-91.23-c1-0-6
Degree $2$
Conductor $91$
Sign $-0.652 + 0.758i$
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  − 2.30i·2-s + (0.736 − 1.27i)3-s − 3.30·4-s + (0.733 + 0.423i)5-s + (−2.93 − 1.69i)6-s + (−0.357 + 2.62i)7-s + 3.00i·8-s + (0.414 + 0.718i)9-s + (0.975 − 1.69i)10-s + (1.30 + 0.751i)11-s + (−2.43 + 4.21i)12-s + (−2.92 − 2.11i)13-s + (6.03 + 0.824i)14-s + (1.08 − 0.624i)15-s + 0.313·16-s − 2.07·17-s + ⋯
L(s)  = 1  − 1.62i·2-s + (0.425 − 0.736i)3-s − 1.65·4-s + (0.328 + 0.189i)5-s + (−1.19 − 0.692i)6-s + (−0.135 + 0.990i)7-s + 1.06i·8-s + (0.138 + 0.239i)9-s + (0.308 − 0.534i)10-s + (0.392 + 0.226i)11-s + (−0.702 + 1.21i)12-s + (−0.810 − 0.585i)13-s + (1.61 + 0.220i)14-s + (0.279 − 0.161i)15-s + 0.0782·16-s − 0.502·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 + 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.652 + 0.758i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.652 + 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.445680 - 0.971294i\)
\(L(\frac12)\) \(\approx\) \(0.445680 - 0.971294i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.357 - 2.62i)T \)
13 \( 1 + (2.92 + 2.11i)T \)
good2 \( 1 + 2.30iT - 2T^{2} \)
3 \( 1 + (-0.736 + 1.27i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.733 - 0.423i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.30 - 0.751i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 2.07T + 17T^{2} \)
19 \( 1 + (-0.0410 + 0.0237i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 7.81T + 23T^{2} \)
29 \( 1 + (0.679 + 1.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.80 - 3.93i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.70iT - 37T^{2} \)
41 \( 1 + (8.67 - 5.00i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.63 + 8.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.311 + 0.180i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.35 + 2.34i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.64iT - 59T^{2} \)
61 \( 1 + (2.26 + 3.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.76 + 1.02i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-12.3 - 7.10i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.85 + 3.38i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.82 - 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.5iT - 83T^{2} \)
89 \( 1 + 17.5iT - 89T^{2} \)
97 \( 1 + (-0.369 - 0.213i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.21098375555296210455720344612, −12.62639870109337654898105823890, −11.74347807521771168101617378413, −10.56195323780077872098810316459, −9.524270248362346884155124418671, −8.511410383031262519802813124368, −6.89838642056512864900644532493, −4.98267779755160012875900137534, −2.98211847019348852409088570769, −1.93215821454881907906339766027, 3.93187325209653416162588145160, 5.08561285753788235415578127806, 6.64402339353704994265419226554, 7.46868908341874741275798635430, 9.028850275180883122244604245973, 9.507970314615427977720799930567, 11.01250540777947229221813818663, 12.90140203380108369188756611121, 13.91526529627736121929986990324, 14.69259040322991413121029658272

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