# Properties

 Label 2-91-91.31-c1-0-6 Degree $2$ Conductor $91$ Sign $-0.0968 + 0.995i$ Analytic cond. $0.726638$ Root an. cond. $0.852431$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−0.746 − 0.200i)2-s + (−0.421 − 0.243i)3-s + (−1.21 − 0.701i)4-s + (0.472 − 1.76i)5-s + (0.266 + 0.266i)6-s + (−0.210 − 2.63i)7-s + (1.85 + 1.85i)8-s + (−1.38 − 2.39i)9-s + (−0.705 + 1.22i)10-s + (0.990 − 0.265i)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 + (−0.527 − 0.141i)2-s + (−0.243 − 0.140i)3-s + (−0.607 − 0.350i)4-s + (0.211 − 0.788i)5-s + (0.108 + 0.108i)6-s + (−0.0796 − 0.996i)7-s + (0.657 + 0.657i)8-s + (−0.460 − 0.797i)9-s + (−0.222 + 0.386i)10-s + (0.298 − 0.0800i)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 & (-0.0968 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0968 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$91$$    =    $$7 \cdot 13$$ Sign: $-0.0968 + 0.995i$ 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: $$(2,\ 91,\ (\ :1/2),\ -0.0968 + 0.995i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.409766 - 0.451577i$$ $$L(\frac12)$$ $$\approx$$ $$0.409766 - 0.451577i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

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

−14.00998077558890050122206823884, −12.71191820220181739684305870597, −11.65133394677447277056633416735, −10.34342127924307845413148614012, −9.378600360207768182485916068368, −8.548656571483139079232375125808, −6.99649301417255582390982921848, −5.45403370585516561370357061800, −4.10062245946867137866254355658, −1.02509958508267018611401192096, 2.93215372098286006262286611485, 4.93673584719689706704388867656, 6.32135201731032512167399312230, 7.88641280324703509084661225926, 8.763156701757725204306879671257, 10.04093586633991189954592313937, 10.90386495849704973061700958326, 12.29915657195273366294204215095, 13.28448265147358339694040831825, 14.45611605116729390029734075522