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Properties

Label 2-91-13.3-c1-0-4
Degree $2$
Conductor $91$
Sign $0.859 + 0.511i$
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  + (−0.190 − 0.330i)2-s + (0.190 + 0.330i)3-s + (0.927 − 1.60i)4-s + 0.381·5-s + (0.0729 − 0.126i)6-s + (−0.5 + 0.866i)7-s − 1.47·8-s + (1.42 − 2.47i)9-s + (−0.0729 − 0.126i)10-s + (2.42 + 4.20i)11-s + 0.708·12-s + (−2.5 − 2.59i)13-s + 0.381·14-s + (0.0729 + 0.126i)15-s + (−1.57 − 2.72i)16-s + (−3.73 + 6.47i)17-s + ⋯
L(s)  = 1  + (−0.135 − 0.233i)2-s + (0.110 + 0.190i)3-s + (0.463 − 0.802i)4-s + 0.170·5-s + (0.0297 − 0.0515i)6-s + (−0.188 + 0.327i)7-s − 0.520·8-s + (0.475 − 0.823i)9-s + (−0.0230 − 0.0399i)10-s + (0.731 + 1.26i)11-s + 0.204·12-s + (−0.693 − 0.720i)13-s + 0.102·14-s + (0.0188 + 0.0326i)15-s + (−0.393 − 0.681i)16-s + (−0.906 + 1.56i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.995207 - 0.273516i\)
\(L(\frac12)\) \(\approx\) \(0.995207 - 0.273516i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (2.5 + 2.59i)T \)
good2 \( 1 + (0.190 + 0.330i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.190 - 0.330i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 0.381T + 5T^{2} \)
11 \( 1 + (-2.42 - 4.20i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.73 - 6.47i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.42 - 4.20i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.23 + 3.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.04 - 3.54i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.70T + 31T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.61 + 4.53i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.78 + 6.54i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.23T + 47T^{2} \)
53 \( 1 - 8.23T + 53T^{2} \)
59 \( 1 + (1.11 - 1.93i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.354 + 0.613i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.09 - 7.08i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 6.70T + 83T^{2} \)
89 \( 1 + (8.04 + 13.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.07 - 10.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.33555255295221552264652711966, −12.56780066415874169587923508063, −12.08007446407490272512428655527, −10.36376984825053983080344994761, −9.962769946607979302328272250088, −8.709274848606384387222421860697, −6.90104654289965789026058436281, −5.95985015030157177617146234250, −4.18520982427647242351655117503, −2.02725007638130017726342416419, 2.61673524597116849145203397389, 4.41789987200615332892665062700, 6.42749491066853244813148334199, 7.32204083940866826485613760626, 8.492197260027017056801967848497, 9.659386416556933473818658366444, 11.26155316790167454706333594506, 11.86699581261852251228486427006, 13.49186148896404470949798318737, 13.74816346448013485998801686248

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