Properties

Label 2-91-91.89-c1-0-1
Degree $2$
Conductor $91$
Sign $0.991 + 0.132i$
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  + (−0.270 − 0.270i)2-s + (0.792 + 0.457i)3-s − 1.85i·4-s + (0.959 + 3.58i)5-s + (−0.0906 − 0.338i)6-s + (1.30 − 2.30i)7-s + (−1.04 + 1.04i)8-s + (−1.08 − 1.87i)9-s + (0.709 − 1.22i)10-s + (0.0226 + 0.0846i)11-s + (0.847 − 1.46i)12-s + (1.63 + 3.21i)13-s + (−0.975 + 0.270i)14-s + (−0.877 + 3.27i)15-s − 3.14·16-s − 5.89·17-s + ⋯
L(s)  = 1  + (−0.191 − 0.191i)2-s + (0.457 + 0.264i)3-s − 0.926i·4-s + (0.429 + 1.60i)5-s + (−0.0369 − 0.138i)6-s + (0.492 − 0.870i)7-s + (−0.368 + 0.368i)8-s + (−0.360 − 0.624i)9-s + (0.224 − 0.388i)10-s + (0.00683 + 0.0255i)11-s + (0.244 − 0.423i)12-s + (0.453 + 0.891i)13-s + (−0.260 + 0.0722i)14-s + (−0.226 + 0.845i)15-s − 0.785·16-s − 1.42·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.991 + 0.132i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.991 + 0.132i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05031 - 0.0696964i\)
\(L(\frac12)\) \(\approx\) \(1.05031 - 0.0696964i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.30 + 2.30i)T \)
13 \( 1 + (-1.63 - 3.21i)T \)
good2 \( 1 + (0.270 + 0.270i)T + 2iT^{2} \)
3 \( 1 + (-0.792 - 0.457i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.959 - 3.58i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.0226 - 0.0846i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + 5.89T + 17T^{2} \)
19 \( 1 + (3.58 + 0.960i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 0.446iT - 23T^{2} \)
29 \( 1 + (-0.706 - 1.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.94 + 0.520i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.87 + 1.87i)T - 37iT^{2} \)
41 \( 1 + (-3.00 - 0.804i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-8.64 - 4.99i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-8.84 + 2.36i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.28 + 10.8i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.05 - 5.05i)T + 59iT^{2} \)
61 \( 1 + (-0.110 + 0.0638i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.61 + 2.57i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (9.83 - 2.63i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.37 - 8.84i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.75 + 3.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.17 - 2.17i)T - 83iT^{2} \)
89 \( 1 + (-1.19 - 1.19i)T + 89iT^{2} \)
97 \( 1 + (0.452 + 1.68i)T + (-84.0 + 48.5i)T^{2} \)
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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

−14.37001372231851102745706275847, −13.49676987209319951550595492361, −11.26427079527801325398690593122, −10.91598523885031378251854790249, −9.844917962265764062367464887490, −8.823762018336910023627932614599, −6.96761696206945074766930396016, −6.19289742412672805439518430923, −4.12545957803680133277103724601, −2.32858484716153012468007403870, 2.32821416543514002204619285075, 4.51030850489490485328337640069, 5.84202755773653771887084347547, 7.80904840325842645420169603488, 8.651359187209868707421902184652, 9.020049592220021760330520368533, 11.04489627413935216977557651730, 12.40122075031619076445587958219, 12.93351960185989138827129122560, 13.80521013445166331030818318820

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