Properties

Label 2-91-91.33-c1-0-3
Degree $2$
Conductor $91$
Sign $0.986 + 0.164i$
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  + (0.470 − 1.75i)2-s + 3.06i·3-s + (−1.12 − 0.650i)4-s + (0.288 + 1.07i)5-s + (5.38 + 1.44i)6-s + (0.381 − 2.61i)7-s + (0.899 − 0.899i)8-s − 6.42·9-s + 2.02·10-s + (−0.0909 + 0.0909i)11-s + (1.99 − 3.45i)12-s + (−3.59 + 0.291i)13-s + (−4.41 − 1.90i)14-s + (−3.30 + 0.884i)15-s + (−2.45 − 4.25i)16-s + (−0.136 + 0.236i)17-s + ⋯
L(s)  = 1  + (0.332 − 1.24i)2-s + 1.77i·3-s + (−0.562 − 0.325i)4-s + (0.128 + 0.480i)5-s + (2.19 + 0.589i)6-s + (0.144 − 0.989i)7-s + (0.317 − 0.317i)8-s − 2.14·9-s + 0.639·10-s + (−0.0274 + 0.0274i)11-s + (0.576 − 0.997i)12-s + (−0.996 + 0.0807i)13-s + (−1.17 − 0.508i)14-s + (−0.852 + 0.228i)15-s + (−0.613 − 1.06i)16-s + (−0.0331 + 0.0574i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.986 + 0.164i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.986 + 0.164i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16633 - 0.0963873i\)
\(L(\frac12)\) \(\approx\) \(1.16633 - 0.0963873i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.381 + 2.61i)T \)
13 \( 1 + (3.59 - 0.291i)T \)
good2 \( 1 + (-0.470 + 1.75i)T + (-1.73 - i)T^{2} \)
3 \( 1 - 3.06iT - 3T^{2} \)
5 \( 1 + (-0.288 - 1.07i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.0909 - 0.0909i)T - 11iT^{2} \)
17 \( 1 + (0.136 - 0.236i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.80 + 2.80i)T - 19iT^{2} \)
23 \( 1 + (0.426 - 0.245i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.62 - 8.00i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.92 + 1.85i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-3.85 - 1.03i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.31 - 8.63i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-8.44 + 4.87i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.66 - 1.51i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.467 - 0.809i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.64 + 1.78i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 - 7.78iT - 61T^{2} \)
67 \( 1 + (0.865 + 0.865i)T + 67iT^{2} \)
71 \( 1 + (0.793 - 2.96i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.0318 + 0.118i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.72 - 9.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.07 + 8.07i)T - 83iT^{2} \)
89 \( 1 + (1.09 - 4.09i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.39 + 0.641i)T + (84.0 + 48.5i)T^{2} \)
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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

−14.20795812770598824310255307585, −12.89260471550167624899296451598, −11.41629542067788813257556935771, −10.83791099605229481873663208034, −10.06491775591257396923433641305, −9.288989387733384041091969188881, −7.26146350958365533814823559514, −5.05025730871849737987731856748, −4.05099392447672875276615350699, −2.93539337621654851204364530679, 2.12661236613785679772904091453, 5.29767831465369580510383395021, 6.05491354925574613157410155446, 7.30448188244732540509032486085, 7.969217202411498067340493483218, 9.133284890312535903283292777887, 11.40754753374275789577560071369, 12.42288756974936006374979347378, 13.11193840350598777191789347447, 14.22772842312170060177534428421

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