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Properties

Label 2-91-91.31-c1-0-3
Degree $2$
Conductor $91$
Sign $-0.832 + 0.553i$
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  + (−2.60 − 0.697i)2-s + (−0.657 − 0.379i)3-s + (4.55 + 2.63i)4-s + (0.654 − 2.44i)5-s + (1.44 + 1.44i)6-s + (−2.54 + 0.722i)7-s + (−6.22 − 6.22i)8-s + (−1.21 − 2.09i)9-s + (−3.40 + 5.89i)10-s + (−2.08 + 0.557i)11-s + (−1.99 − 3.46i)12-s + (−1.44 − 3.30i)13-s + (7.13 − 0.104i)14-s + (−1.35 + 1.35i)15-s + (6.59 + 11.4i)16-s + (0.700 − 1.21i)17-s + ⋯
L(s)  = 1  + (−1.84 − 0.493i)2-s + (−0.379 − 0.219i)3-s + (2.27 + 1.31i)4-s + (0.292 − 1.09i)5-s + (0.590 + 0.590i)6-s + (−0.962 + 0.272i)7-s + (−2.19 − 2.19i)8-s + (−0.403 − 0.699i)9-s + (−1.07 + 1.86i)10-s + (−0.627 + 0.168i)11-s + (−0.576 − 0.999i)12-s + (−0.401 − 0.915i)13-s + (1.90 − 0.0279i)14-s + (−0.350 + 0.350i)15-s + (1.64 + 2.85i)16-s + (0.169 − 0.294i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.832 + 0.553i$
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: \((2,\ 91,\ (\ :1/2),\ -0.832 + 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0813248 - 0.269280i\)
\(L(\frac12)\) \(\approx\) \(0.0813248 - 0.269280i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.54 - 0.722i)T \)
13 \( 1 + (1.44 + 3.30i)T \)
good2 \( 1 + (2.60 + 0.697i)T + (1.73 + i)T^{2} \)
3 \( 1 + (0.657 + 0.379i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.654 + 2.44i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.08 - 0.557i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.700 + 1.21i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.541 + 2.02i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.13 + 0.657i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.56T + 29T^{2} \)
31 \( 1 + (-7.03 + 1.88i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.591 + 2.20i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.69 - 2.69i)T + 41iT^{2} \)
43 \( 1 + 0.437iT - 43T^{2} \)
47 \( 1 + (-7.74 - 2.07i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.26 + 2.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.02 + 7.54i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-6.57 + 3.79i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.146 - 0.548i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (10.7 - 10.7i)T - 71iT^{2} \)
73 \( 1 + (-3.18 - 11.8i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (7.19 + 12.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.82 - 3.82i)T + 83iT^{2} \)
89 \( 1 + (0.0501 + 0.0134i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (9.43 + 9.43i)T + 97iT^{2} \)
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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

−12.92800048211682028076530154877, −12.43441271851819711775692669779, −11.35770187378939067704538991672, −10.04294251458327244985530496147, −9.335546229458197928226784354173, −8.430863056903688669372157414028, −7.14043775313364900571794926116, −5.75517904701223732558950367943, −2.83522810593315652353081890689, −0.60251546054185604687340289240, 2.57764893822312737181679365894, 5.84117297383455210425303238358, 6.79643606808039043342769300101, 7.76243040651236617006383755859, 9.165534144650775278910508536517, 10.28308144595992347669514797858, 10.62041588163618566128132588993, 11.79881011207988419675388495904, 13.78344912818234545456658753171, 14.92580713011005741041413125363

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