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Properties

Label 2-91-91.31-c1-0-1
Degree $2$
Conductor $91$
Sign $0.631 - 0.775i$
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  + (−1.84 − 0.493i)2-s + (2.29 + 1.32i)3-s + (1.41 + 0.818i)4-s + (−0.885 + 3.30i)5-s + (−3.57 − 3.57i)6-s + (−2.39 − 1.12i)7-s + (0.489 + 0.489i)8-s + (2.00 + 3.47i)9-s + (3.26 − 5.65i)10-s + (1.66 − 0.445i)11-s + (2.16 + 3.75i)12-s + (3.57 + 0.501i)13-s + (3.84 + 3.26i)14-s + (−6.40 + 6.40i)15-s + (−2.29 − 3.97i)16-s + (1.22 − 2.12i)17-s + ⋯
L(s)  = 1  + (−1.30 − 0.349i)2-s + (1.32 + 0.764i)3-s + (0.708 + 0.409i)4-s + (−0.396 + 1.47i)5-s + (−1.45 − 1.45i)6-s + (−0.904 − 0.426i)7-s + (0.173 + 0.173i)8-s + (0.668 + 1.15i)9-s + (1.03 − 1.78i)10-s + (0.501 − 0.134i)11-s + (0.625 + 1.08i)12-s + (0.990 + 0.138i)13-s + (1.02 + 0.871i)14-s + (−1.65 + 1.65i)15-s + (−0.574 − 0.994i)16-s + (0.297 − 0.515i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.631 - 0.775i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.631 - 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.640855 + 0.304640i\)
\(L(\frac12)\) \(\approx\) \(0.640855 + 0.304640i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.39 + 1.12i)T \)
13 \( 1 + (-3.57 - 0.501i)T \)
good2 \( 1 + (1.84 + 0.493i)T + (1.73 + i)T^{2} \)
3 \( 1 + (-2.29 - 1.32i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.885 - 3.30i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.66 + 0.445i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.22 + 2.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.34 + 5.03i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.97 - 2.29i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.184T + 29T^{2} \)
31 \( 1 + (-2.46 + 0.659i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.0563 + 0.210i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (4.63 + 4.63i)T + 41iT^{2} \)
43 \( 1 + 0.562iT - 43T^{2} \)
47 \( 1 + (-3.72 - 0.998i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.67 - 4.63i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.73 + 13.9i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.30 - 0.754i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.78 - 6.67i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.70 + 1.70i)T - 71iT^{2} \)
73 \( 1 + (-3.15 - 11.7i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.48 + 2.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.504 + 0.504i)T + 83iT^{2} \)
89 \( 1 + (7.20 + 1.92i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-12.0 - 12.0i)T + 97iT^{2} \)
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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

−14.16703069331572154684034281696, −13.71432605415757206371456734285, −11.49431569167913148024278678853, −10.58024506336252560452173829129, −9.772202258595065069503497242779, −9.005441531349650294504314336264, −7.82697243331975954079043593118, −6.77278908190088737155518195305, −3.79122278160912159262334938036, −2.78345594864402412367436934665, 1.41696384238265858793824449392, 3.79412718349241956775227682476, 6.28581911857875927407311130232, 7.78280803494215577077249323333, 8.467076403211925105049402604696, 9.031420117371498385424188382883, 10.01619188671402570221206753253, 12.15259931681870365135736262731, 12.89810287019203408686143267588, 13.78691398648521198043141269116

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