Properties

Label 2-91-91.20-c1-0-7
Degree $2$
Conductor $91$
Sign $-0.997 - 0.0697i$
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  + (−0.112 + 0.0302i)2-s + (−2.25 − 1.29i)3-s + (−1.72 + 0.993i)4-s + (−1.24 + 1.24i)5-s + (0.293 + 0.0785i)6-s + (−2.18 − 1.49i)7-s + (0.329 − 0.329i)8-s + (1.87 + 3.25i)9-s + (0.103 − 0.178i)10-s + (−0.506 − 1.89i)11-s + 5.16·12-s + (−1.85 + 3.09i)13-s + (0.291 + 0.102i)14-s + (4.43 − 1.18i)15-s + (1.95 − 3.39i)16-s + (−2.13 − 3.70i)17-s + ⋯
L(s)  = 1  + (−0.0797 + 0.0213i)2-s + (−1.29 − 0.750i)3-s + (−0.860 + 0.496i)4-s + (−0.558 + 0.558i)5-s + (0.119 + 0.0320i)6-s + (−0.824 − 0.565i)7-s + (0.116 − 0.116i)8-s + (0.625 + 1.08i)9-s + (0.0325 − 0.0564i)10-s + (−0.152 − 0.570i)11-s + 1.49·12-s + (−0.515 + 0.857i)13-s + (0.0778 + 0.0275i)14-s + (1.14 − 0.306i)15-s + (0.489 − 0.848i)16-s + (−0.518 − 0.898i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.997 - 0.0697i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.997 - 0.0697i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00123326 + 0.0353182i\)
\(L(\frac12)\) \(\approx\) \(0.00123326 + 0.0353182i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.18 + 1.49i)T \)
13 \( 1 + (1.85 - 3.09i)T \)
good2 \( 1 + (0.112 - 0.0302i)T + (1.73 - i)T^{2} \)
3 \( 1 + (2.25 + 1.29i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.24 - 1.24i)T - 5iT^{2} \)
11 \( 1 + (0.506 + 1.89i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.13 + 3.70i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.12 - 1.10i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (5.53 + 3.19i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.57 - 6.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.02 - 3.02i)T - 31iT^{2} \)
37 \( 1 + (0.732 + 2.73i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.94 + 11.0i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.55 - 0.896i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.68 + 4.68i)T + 47iT^{2} \)
53 \( 1 - 4.27T + 53T^{2} \)
59 \( 1 + (0.436 - 1.62i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.66 + 1.54i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.0190 - 0.00510i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.23 + 4.59i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.698 - 0.698i)T + 73iT^{2} \)
79 \( 1 + 5.93T + 79T^{2} \)
83 \( 1 + (-9.87 + 9.87i)T - 83iT^{2} \)
89 \( 1 + (7.76 - 2.07i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-14.2 - 3.82i)T + (84.0 + 48.5i)T^{2} \)
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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

−13.40222234411206017482013776963, −12.36205601473699206978505667366, −11.63599832054258260409702821859, −10.49893907789993754756888064991, −9.155687019775643967428133926740, −7.45551111067986559680335188635, −6.82329186014411233572439745503, −5.26970707873687033402887584390, −3.64181158332693580763834498962, −0.05083097366308138595948967712, 4.09986853610558038249842788853, 5.19701777486391296568328249832, 6.07937944219011133221049449403, 8.104887248074270336211672847819, 9.630505854871371891236252141258, 10.06334664925066855293372507377, 11.45260132350960212206523077206, 12.41509216778766429382524952269, 13.28883932508923685931319061928, 15.06297599993898813790524470895

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