Properties

Label 2-91-91.5-c1-0-3
Degree $2$
Conductor $91$
Sign $0.859 - 0.511i$
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  + (0.200 + 0.746i)2-s + (−0.421 + 0.243i)3-s + (1.21 − 0.701i)4-s + (1.76 − 0.472i)5-s + (−0.266 − 0.266i)6-s + (−2.63 − 0.210i)7-s + (1.85 + 1.85i)8-s + (−1.38 + 2.39i)9-s + (0.705 + 1.22i)10-s + (−0.265 + 0.990i)11-s + (−0.341 + 0.591i)12-s + (0.266 − 3.59i)13-s + (−0.370 − 2.01i)14-s + (−0.628 + 0.628i)15-s + (0.386 − 0.669i)16-s + (−2.60 − 4.50i)17-s + ⋯
L(s)  = 1  + (0.141 + 0.527i)2-s + (−0.243 + 0.140i)3-s + (0.607 − 0.350i)4-s + (0.788 − 0.211i)5-s + (−0.108 − 0.108i)6-s + (−0.996 − 0.0796i)7-s + (0.657 + 0.657i)8-s + (−0.460 + 0.797i)9-s + (0.222 + 0.386i)10-s + (−0.0800 + 0.298i)11-s + (−0.0986 + 0.170i)12-s + (0.0738 − 0.997i)13-s + (−0.0989 − 0.537i)14-s + (−0.162 + 0.162i)15-s + (0.0966 − 0.167i)16-s + (−0.630 − 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08280 + 0.298047i\)
\(L(\frac12)\) \(\approx\) \(1.08280 + 0.298047i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.63 + 0.210i)T \)
13 \( 1 + (-0.266 + 3.59i)T \)
good2 \( 1 + (-0.200 - 0.746i)T + (-1.73 + i)T^{2} \)
3 \( 1 + (0.421 - 0.243i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.76 + 0.472i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.265 - 0.990i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.60 + 4.50i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.07 - 1.36i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.730 + 0.421i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 10.3T + 29T^{2} \)
31 \( 1 + (1.52 - 5.69i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (6.03 - 1.61i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.0927 - 0.0927i)T + 41iT^{2} \)
43 \( 1 + 7.36iT - 43T^{2} \)
47 \( 1 + (-0.583 - 2.17i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.38 - 5.86i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-9.73 - 2.60i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.13 - 0.653i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.16 + 1.11i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (6.02 - 6.02i)T - 71iT^{2} \)
73 \( 1 + (-10.9 - 2.93i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.16 + 8.94i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.16 - 4.16i)T + 83iT^{2} \)
89 \( 1 + (2.00 + 7.49i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.49 + 2.49i)T + 97iT^{2} \)
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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

−14.10246641052522447521964913105, −13.38519057734006804942113522449, −12.14167520710581128478868388456, −10.65349194779890084344401058641, −10.11755294338184767724558284001, −8.560039592441857401005302650076, −7.05492573591506679168223116522, −6.01443678984491001750504314978, −5.03801951681084398793312731173, −2.51920221749349924822644345542, 2.32787128841555273065405676477, 3.85005019200505507332980299425, 6.27896098530151871928974973632, 6.59968093505630890600030514303, 8.616769796651526886807658496274, 9.860180435963222412964142751644, 10.86574359813465614258321536885, 11.94335485210354568328618681122, 12.81237891104338053114462727720, 13.69751931566068923879339614372

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