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Properties

Label 2-91-91.31-c1-0-2
Degree $2$
Conductor $91$
Sign $0.907 - 0.421i$
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  + (0.694 + 0.186i)2-s + (1.44 + 0.832i)3-s + (−1.28 − 0.741i)4-s + (−0.501 + 1.87i)5-s + (0.846 + 0.846i)6-s + (2.52 − 0.783i)7-s + (−1.77 − 1.77i)8-s + (−0.115 − 0.199i)9-s + (−0.696 + 1.20i)10-s + (−3.08 + 0.825i)11-s + (−1.23 − 2.13i)12-s + (−0.846 − 3.50i)13-s + (1.90 − 0.0737i)14-s + (−2.27 + 2.27i)15-s + (0.582 + 1.00i)16-s + (0.254 − 0.440i)17-s + ⋯
L(s)  = 1  + (0.491 + 0.131i)2-s + (0.832 + 0.480i)3-s + (−0.642 − 0.370i)4-s + (−0.224 + 0.836i)5-s + (0.345 + 0.345i)6-s + (0.955 − 0.296i)7-s + (−0.626 − 0.626i)8-s + (−0.0383 − 0.0664i)9-s + (−0.220 + 0.381i)10-s + (−0.929 + 0.248i)11-s + (−0.356 − 0.617i)12-s + (−0.234 − 0.972i)13-s + (0.508 − 0.0196i)14-s + (−0.588 + 0.588i)15-s + (0.145 + 0.252i)16-s + (0.0616 − 0.106i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 - 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.907 - 0.421i$
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: \((2,\ 91,\ (\ :1/2),\ 0.907 - 0.421i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27661 + 0.281851i\)
\(L(\frac12)\) \(\approx\) \(1.27661 + 0.281851i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.52 + 0.783i)T \)
13 \( 1 + (0.846 + 3.50i)T \)
good2 \( 1 + (-0.694 - 0.186i)T + (1.73 + i)T^{2} \)
3 \( 1 + (-1.44 - 0.832i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.501 - 1.87i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (3.08 - 0.825i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.254 + 0.440i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.710 - 2.65i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.49 - 1.44i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.40T + 29T^{2} \)
31 \( 1 + (-3.08 + 0.827i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.51 - 9.40i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-5.34 - 5.34i)T + 41iT^{2} \)
43 \( 1 + 12.5iT - 43T^{2} \)
47 \( 1 + (-10.7 - 2.88i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.42 - 5.93i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.00 - 3.74i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (5.51 - 3.18i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.75 + 6.56i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.90 - 1.90i)T - 71iT^{2} \)
73 \( 1 + (-0.0676 - 0.252i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.78 + 4.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.86 + 5.86i)T + 83iT^{2} \)
89 \( 1 + (11.8 + 3.17i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-7.04 - 7.04i)T + 97iT^{2} \)
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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

−14.25755557257259086661810076813, −13.55910156945593427572389587997, −12.20606474939153521268607255919, −10.66127633989146028832459084793, −9.940601276293743322531054341953, −8.552401743150662331982408675173, −7.54586209286695784093195415485, −5.71378156622031214538464127076, −4.37339223923786810030255996710, −3.05401444414797194066344773993, 2.43390796304407425354939976551, 4.33319018602057818934418045861, 5.34811634161643745161295116312, 7.64243135287278316264483798715, 8.451545758799155146547404433586, 9.124115107300034207811765623299, 11.09189985561210378126578594251, 12.27711122023439142320445196686, 13.02732537260827505193556251536, 13.98741214670197224173223986515

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