Properties

Label 2-91-91.20-c1-0-1
Degree $2$
Conductor $91$
Sign $-0.106 - 0.994i$
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  + (−0.892 + 0.239i)2-s + (0.988 + 0.570i)3-s + (−0.993 + 0.573i)4-s + (−2.80 + 2.80i)5-s + (−1.01 − 0.272i)6-s + (2.12 + 1.57i)7-s + (2.05 − 2.05i)8-s + (−0.848 − 1.47i)9-s + (1.82 − 3.16i)10-s + (0.544 + 2.03i)11-s − 1.30·12-s + (3.41 + 1.16i)13-s + (−2.27 − 0.893i)14-s + (−4.36 + 1.16i)15-s + (−0.195 + 0.339i)16-s + (−1.58 − 2.74i)17-s + ⋯
L(s)  = 1  + (−0.630 + 0.169i)2-s + (0.570 + 0.329i)3-s + (−0.496 + 0.286i)4-s + (−1.25 + 1.25i)5-s + (−0.415 − 0.111i)6-s + (0.804 + 0.594i)7-s + (0.726 − 0.726i)8-s + (−0.282 − 0.490i)9-s + (0.578 − 1.00i)10-s + (0.164 + 0.613i)11-s − 0.377·12-s + (0.946 + 0.324i)13-s + (−0.607 − 0.238i)14-s + (−1.12 + 0.301i)15-s + (−0.0489 + 0.0847i)16-s + (−0.384 − 0.666i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.106 - 0.994i$
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: \((2,\ 91,\ (\ :1/2),\ -0.106 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.451196 + 0.502122i\)
\(L(\frac12)\) \(\approx\) \(0.451196 + 0.502122i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.12 - 1.57i)T \)
13 \( 1 + (-3.41 - 1.16i)T \)
good2 \( 1 + (0.892 - 0.239i)T + (1.73 - i)T^{2} \)
3 \( 1 + (-0.988 - 0.570i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.80 - 2.80i)T - 5iT^{2} \)
11 \( 1 + (-0.544 - 2.03i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.58 + 2.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.12 - 0.302i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.26 - 1.88i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.584 - 1.01i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.57 + 3.57i)T - 31iT^{2} \)
37 \( 1 + (-1.14 - 4.26i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.85 - 6.93i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.91 - 1.10i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.21 + 8.21i)T + 47iT^{2} \)
53 \( 1 - 4.89T + 53T^{2} \)
59 \( 1 + (-0.0633 + 0.236i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-10.7 + 6.18i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.95 - 2.66i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.60 + 5.98i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (3.47 + 3.47i)T + 73iT^{2} \)
79 \( 1 + 2.69T + 79T^{2} \)
83 \( 1 + (-3.31 + 3.31i)T - 83iT^{2} \)
89 \( 1 + (-6.71 + 1.80i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (15.8 + 4.24i)T + (84.0 + 48.5i)T^{2} \)
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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

−14.72277160682044266792861977875, −13.57314012504888595994465291129, −11.89363475128794373792762122533, −11.22760725361810793900683895499, −9.788696978346026720041775543940, −8.684297712699076232643435670821, −7.908197013967134849294754436797, −6.77891265553642557501566193755, −4.39804818711276359345084503789, −3.21567724563409809160966855716, 1.12940559217859421192139152019, 4.02718190754186313894289292193, 5.19836774581948512780317372565, 7.66459889030041382708701934140, 8.454693113769158443902825150573, 8.809065611643694340022670280504, 10.69846929470157872985055912139, 11.42837040235937696487839974410, 12.94350697615185947179972845011, 13.69938874575319452330240128018

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