# Properties

 Label 2-91-91.34-c1-0-3 Degree $2$ Conductor $91$ Sign $0.950 + 0.312i$ 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 + (0.854 + 0.854i)2-s − 2.27i·3-s − 0.539i·4-s + (−0.612 + 0.612i)5-s + (1.94 − 1.94i)6-s + (0.0148 + 2.64i)7-s + (2.17 − 2.17i)8-s − 2.17·9-s − 1.04·10-s + (−1.85 + 1.85i)11-s − 1.22·12-s + (−0.104 + 3.60i)13-s + (−2.24 + 2.27i)14-s + (1.39 + 1.39i)15-s + 2.63·16-s − 3.04·17-s + ⋯
 L(s)  = 1 + (0.604 + 0.604i)2-s − 1.31i·3-s − 0.269i·4-s + (−0.274 + 0.274i)5-s + (0.793 − 0.793i)6-s + (0.00559 + 0.999i)7-s + (0.767 − 0.767i)8-s − 0.723·9-s − 0.331·10-s + (−0.559 + 0.559i)11-s − 0.353·12-s + (−0.0289 + 0.999i)13-s + (−0.600 + 0.607i)14-s + (0.359 + 0.359i)15-s + 0.657·16-s − 0.739·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$91$$    =    $$7 \cdot 13$$ Sign: $0.950 + 0.312i$ Analytic conductor: $$0.726638$$ Root analytic conductor: $$0.852431$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{91} (34, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 91,\ (\ :1/2),\ 0.950 + 0.312i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.23642 - 0.197837i$$ $$L(\frac12)$$ $$\approx$$ $$1.23642 - 0.197837i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1 + (-0.0148 - 2.64i)T$$
13 $$1 + (0.104 - 3.60i)T$$
good2 $$1 + (-0.854 - 0.854i)T + 2iT^{2}$$
3 $$1 + 2.27iT - 3T^{2}$$
5 $$1 + (0.612 - 0.612i)T - 5iT^{2}$$
11 $$1 + (1.85 - 1.85i)T - 11iT^{2}$$
17 $$1 + 3.04T + 17T^{2}$$
19 $$1 + (-0.104 + 0.104i)T - 19iT^{2}$$
23 $$1 + 6.51iT - 23T^{2}$$
29 $$1 + 3.78T + 29T^{2}$$
31 $$1 + (-6.77 + 6.77i)T - 31iT^{2}$$
37 $$1 + (2.02 - 2.02i)T - 37iT^{2}$$
41 $$1 + (2.27 - 2.27i)T - 41iT^{2}$$
43 $$1 - 3.18iT - 43T^{2}$$
47 $$1 + (-5.21 - 5.21i)T + 47iT^{2}$$
53 $$1 - 3.43T + 53T^{2}$$
59 $$1 + (9.15 + 9.15i)T + 59iT^{2}$$
61 $$1 + 9.20iT - 61T^{2}$$
67 $$1 + (1.04 + 1.04i)T + 67iT^{2}$$
71 $$1 + (4.10 + 4.10i)T + 71iT^{2}$$
73 $$1 + (-6.92 - 6.92i)T + 73iT^{2}$$
79 $$1 - 17.5T + 79T^{2}$$
83 $$1 + (10.5 - 10.5i)T - 83iT^{2}$$
89 $$1 + (-3.39 - 3.39i)T + 89iT^{2}$$
97 $$1 + (-4.44 + 4.44i)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

−13.98353863623808270418021785208, −13.08557002492867303398026498372, −12.28213651575616731888089319413, −11.12984727260103900263187170457, −9.525905642689875561063571596545, −8.014731466645500010289390121330, −6.89225516181246971305928376568, −6.16778644484101142841447054422, −4.67499310526260694379185632790, −2.14438079310811671800561183369, 3.25314211905350525024208082722, 4.21483089158316522589752718223, 5.26248747738750301828740107458, 7.52945371402702319482674317911, 8.709287153944987408213217682562, 10.31152399380200435213994701471, 10.75821816629813635791126614853, 11.93387762342283255854405363266, 13.20332476640183840223656541137, 13.88071921399545743357128352004