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Properties

Label 2-91-91.45-c1-0-1
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: $\chi_{91} (45, \cdot )$
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

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

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