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Properties

Label 2-91-91.73-c1-0-2
Degree $2$
Conductor $91$
Sign $0.943 - 0.330i$
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  + (−0.211 + 0.788i)2-s + (−2.60 − 1.50i)3-s + (1.15 + 0.666i)4-s + (3.03 + 0.814i)5-s + (1.73 − 1.73i)6-s + (2.28 − 1.32i)7-s + (−1.92 + 1.92i)8-s + (3.02 + 5.23i)9-s + (−1.28 + 2.22i)10-s + (−0.131 − 0.491i)11-s + (−2.00 − 3.47i)12-s + (−1.73 − 3.15i)13-s + (0.562 + 2.08i)14-s + (−6.68 − 6.68i)15-s + (0.221 + 0.382i)16-s + (−0.606 + 1.05i)17-s + ⋯
L(s)  = 1  + (−0.149 + 0.557i)2-s + (−1.50 − 0.868i)3-s + (0.577 + 0.333i)4-s + (1.35 + 0.364i)5-s + (0.709 − 0.709i)6-s + (0.865 − 0.501i)7-s + (−0.680 + 0.680i)8-s + (1.00 + 1.74i)9-s + (−0.406 + 0.703i)10-s + (−0.0396 − 0.148i)11-s + (−0.578 − 1.00i)12-s + (−0.481 − 0.876i)13-s + (0.150 + 0.557i)14-s + (−1.72 − 1.72i)15-s + (0.0552 + 0.0957i)16-s + (−0.147 + 0.254i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.943 - 0.330i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.943 - 0.330i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.833408 + 0.141689i\)
\(L(\frac12)\) \(\approx\) \(0.833408 + 0.141689i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.28 + 1.32i)T \)
13 \( 1 + (1.73 + 3.15i)T \)
good2 \( 1 + (0.211 - 0.788i)T + (-1.73 - i)T^{2} \)
3 \( 1 + (2.60 + 1.50i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-3.03 - 0.814i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.131 + 0.491i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.606 - 1.05i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.72 + 0.461i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.51 - 2.60i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.64T + 29T^{2} \)
31 \( 1 + (0.976 + 3.64i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.66 + 0.715i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (5.55 - 5.55i)T - 41iT^{2} \)
43 \( 1 + 7.46iT - 43T^{2} \)
47 \( 1 + (1.26 - 4.73i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-4.30 + 7.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.41 - 0.648i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (9.09 - 5.25i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.15 + 1.91i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.840 - 0.840i)T + 71iT^{2} \)
73 \( 1 + (-2.36 + 0.632i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-6.20 - 10.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.31 - 7.31i)T - 83iT^{2} \)
89 \( 1 + (-2.52 + 9.42i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.93 - 2.93i)T - 97iT^{2} \)
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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

−14.03332328922764730777247046850, −13.02225989671632017298684397174, −11.95648684574443847945642405071, −11.00923196807389462601064750551, −10.18930241547361810261443077358, −8.063472557372703242836380781488, −7.04510063479038354219627162234, −6.10263649494782307455398310019, −5.31373402400755897253505669954, −1.97621729573703326499876079631, 1.88349709835854244249155355006, 4.71753483172813090756385321017, 5.69319092565507202257874929869, 6.59513799237868926937137611446, 9.125567853128504683147587784941, 10.02825281909435632971842857493, 10.74171497651278736253353466360, 11.77257876467299616593218807835, 12.38633762453552799998426305143, 14.12715090578676399704429797280

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