Properties

Label 2-91-91.41-c1-0-2
Degree $2$
Conductor $91$
Sign $0.271 + 0.962i$
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  + (−2.47 − 0.664i)2-s + (−2.33 + 1.34i)3-s + (3.97 + 2.29i)4-s + (−0.281 − 0.281i)5-s + (6.68 − 1.79i)6-s + (−0.201 − 2.63i)7-s + (−4.70 − 4.70i)8-s + (2.13 − 3.69i)9-s + (0.511 + 0.885i)10-s + (0.939 − 3.50i)11-s − 12.3·12-s + (3.44 + 1.06i)13-s + (−1.25 + 6.67i)14-s + (1.03 + 0.277i)15-s + (3.95 + 6.84i)16-s + (2.04 − 3.54i)17-s + ⋯
L(s)  = 1  + (−1.75 − 0.469i)2-s + (−1.34 + 0.778i)3-s + (1.98 + 1.14i)4-s + (−0.125 − 0.125i)5-s + (2.72 − 0.731i)6-s + (−0.0762 − 0.997i)7-s + (−1.66 − 1.66i)8-s + (0.711 − 1.23i)9-s + (0.161 + 0.280i)10-s + (0.283 − 1.05i)11-s − 3.57·12-s + (0.955 + 0.294i)13-s + (−0.334 + 1.78i)14-s + (0.267 + 0.0717i)15-s + (0.987 + 1.71i)16-s + (0.496 − 0.860i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.271 + 0.962i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.271 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.211409 - 0.159968i\)
\(L(\frac12)\) \(\approx\) \(0.211409 - 0.159968i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.201 + 2.63i)T \)
13 \( 1 + (-3.44 - 1.06i)T \)
good2 \( 1 + (2.47 + 0.664i)T + (1.73 + i)T^{2} \)
3 \( 1 + (2.33 - 1.34i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.281 + 0.281i)T + 5iT^{2} \)
11 \( 1 + (-0.939 + 3.50i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.04 + 3.54i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.777 + 0.208i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (4.41 - 2.54i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.00 - 1.74i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.44 + 4.44i)T + 31iT^{2} \)
37 \( 1 + (-0.463 + 1.73i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.578 + 2.15i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.65 + 1.53i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.99 + 5.99i)T - 47iT^{2} \)
53 \( 1 - 9.09T + 53T^{2} \)
59 \( 1 + (-1.92 - 7.17i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.40 - 1.38i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.85 + 1.30i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.582 + 2.17i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (3.50 - 3.50i)T - 73iT^{2} \)
79 \( 1 + 3.61T + 79T^{2} \)
83 \( 1 + (-5.36 - 5.36i)T + 83iT^{2} \)
89 \( 1 + (11.5 + 3.09i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (7.71 - 2.06i)T + (84.0 - 48.5i)T^{2} \)
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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

−13.77907857433001419092204033539, −11.96015417482872792041805758248, −11.32437363717416910052781257599, −10.57298152495110894818964834640, −9.803238659822800104817668154080, −8.606071694182691849422884525544, −7.22454727977435344152395567589, −5.92264203592299581365821186747, −3.83105175680969102623252915648, −0.70185231427497952996475074318, 1.61441421437996250762728169136, 5.67282568787218822129103138379, 6.46254752818505259102035904807, 7.52928619440171145337588151698, 8.661948440820426965308153512231, 9.954554373756480914244718751870, 10.96508151217778519651397616163, 11.87072520786540614395082968481, 12.73483277039187037011100775053, 14.85272682526620287325950075536

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