Properties

Label 2-91-91.89-c1-0-5
Degree $2$
Conductor $91$
Sign $0.0898 + 0.995i$
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  + (0.490 + 0.490i)2-s + (−2.71 − 1.56i)3-s − 1.51i·4-s + (−0.00962 − 0.0359i)5-s + (−0.562 − 2.09i)6-s + (−0.176 − 2.63i)7-s + (1.72 − 1.72i)8-s + (3.39 + 5.88i)9-s + (0.0129 − 0.0223i)10-s + (0.292 + 1.09i)11-s + (−2.37 + 4.11i)12-s + (−3.58 + 0.400i)13-s + (1.20 − 1.38i)14-s + (−0.0301 + 0.112i)15-s − 1.33·16-s + 6.40·17-s + ⋯
L(s)  = 1  + (0.347 + 0.347i)2-s + (−1.56 − 0.903i)3-s − 0.758i·4-s + (−0.00430 − 0.0160i)5-s + (−0.229 − 0.857i)6-s + (−0.0668 − 0.997i)7-s + (0.610 − 0.610i)8-s + (1.13 + 1.96i)9-s + (0.00408 − 0.00707i)10-s + (0.0880 + 0.328i)11-s + (−0.685 + 1.18i)12-s + (−0.993 + 0.111i)13-s + (0.323 − 0.369i)14-s + (−0.00778 + 0.0290i)15-s − 0.334·16-s + 1.55·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0898 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0898 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.0898 + 0.995i$
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: \((2,\ 91,\ (\ :1/2),\ 0.0898 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.542618 - 0.495864i\)
\(L(\frac12)\) \(\approx\) \(0.542618 - 0.495864i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.176 + 2.63i)T \)
13 \( 1 + (3.58 - 0.400i)T \)
good2 \( 1 + (-0.490 - 0.490i)T + 2iT^{2} \)
3 \( 1 + (2.71 + 1.56i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.00962 + 0.0359i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.292 - 1.09i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 - 6.40T + 17T^{2} \)
19 \( 1 + (-3.56 - 0.954i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 2.79iT - 23T^{2} \)
29 \( 1 + (-1.84 - 3.20i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.63 + 0.706i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (3.94 - 3.94i)T - 37iT^{2} \)
41 \( 1 + (0.188 + 0.0505i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.84 - 1.06i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.43 + 1.45i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.295 + 0.512i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.97 - 7.97i)T + 59iT^{2} \)
61 \( 1 + (1.18 - 0.686i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.28 - 1.95i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (9.88 - 2.64i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.707 + 2.64i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.63 - 2.82i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.92 - 4.92i)T - 83iT^{2} \)
89 \( 1 + (-11.9 - 11.9i)T + 89iT^{2} \)
97 \( 1 + (0.663 + 2.47i)T + (-84.0 + 48.5i)T^{2} \)
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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

−13.82240406555672962259073799716, −12.68453381851585815191457970447, −11.89453155874104120780893004190, −10.59079594937459697667232659166, −10.01592320312627895577816765703, −7.45532528751596622188469442647, −6.86275591328844840497299718321, −5.63562418669296886962980025889, −4.70453542639789961054445527797, −1.09140236145934705729979202552, 3.33026048777823887795483945845, 4.92273350062213440018997741729, 5.70653440790462237455040075006, 7.39277794018759767519132152971, 9.150941136516170362768177543983, 10.26085836978094133405559097072, 11.47266379375299181791544576690, 12.01447186658257314590787278691, 12.72702677028491266563321343560, 14.43401231552056072664369824133

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