# Properties

 Label 2-91-91.73-c1-0-4 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 $0$

# Related objects

## 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: $\chi_{91} (73, \cdot )$ 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

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