# Properties

 Label 2-91-13.9-c1-0-2 Degree $2$ Conductor $91$ Sign $0.583 - 0.811i$ 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 + (0.355 − 0.615i)2-s + (−1.20 + 2.08i)3-s + (0.747 + 1.29i)4-s − 1.28·5-s + (0.855 + 1.48i)6-s + (0.5 + 0.866i)7-s + 2.48·8-s + (−1.39 − 2.42i)9-s + (−0.458 + 0.793i)10-s + (1.20 − 2.08i)11-s − 3.60·12-s + (1.25 − 3.38i)13-s + 0.710·14-s + (1.55 − 2.68i)15-s + (−0.613 + 1.06i)16-s + (−1.95 − 3.38i)17-s + ⋯
 L(s)  = 1 + (0.251 − 0.434i)2-s + (−0.695 + 1.20i)3-s + (0.373 + 0.647i)4-s − 0.576·5-s + (0.349 + 0.604i)6-s + (0.188 + 0.327i)7-s + 0.877·8-s + (−0.466 − 0.807i)9-s + (−0.144 + 0.250i)10-s + (0.363 − 0.628i)11-s − 1.03·12-s + (0.347 − 0.937i)13-s + 0.189·14-s + (0.400 − 0.694i)15-s + (−0.153 + 0.265i)16-s + (−0.473 − 0.819i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 - 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.583 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$91$$    =    $$7 \cdot 13$$ Sign: $0.583 - 0.811i$ Analytic conductor: $$0.726638$$ Root analytic conductor: $$0.852431$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{91} (22, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 91,\ (\ :1/2),\ 0.583 - 0.811i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.854637 + 0.438078i$$ $$L(\frac12)$$ $$\approx$$ $$0.854637 + 0.438078i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1 + (-0.5 - 0.866i)T$$
13 $$1 + (-1.25 + 3.38i)T$$
good2 $$1 + (-0.355 + 0.615i)T + (-1 - 1.73i)T^{2}$$
3 $$1 + (1.20 - 2.08i)T + (-1.5 - 2.59i)T^{2}$$
5 $$1 + 1.28T + 5T^{2}$$
11 $$1 + (-1.20 + 2.08i)T + (-5.5 - 9.52i)T^{2}$$
17 $$1 + (1.95 + 3.38i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (-2.94 - 5.10i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (-3.16 + 5.47i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (2.80 - 4.86i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 - 2.20T + 31T^{2}$$
37 $$1 + (-2.55 + 4.43i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + (-3.89 + 6.74i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (0.144 + 0.250i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 - 1.27T + 47T^{2}$$
53 $$1 + 13.6T + 53T^{2}$$
59 $$1 + (-2.01 - 3.49i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-2.30 - 3.98i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (3.78 - 6.56i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + (3.61 + 6.25i)T + (-35.5 + 61.4i)T^{2}$$
73 $$1 + 15.0T + 73T^{2}$$
79 $$1 + 9.30T + 79T^{2}$$
83 $$1 + 1.36T + 83T^{2}$$
89 $$1 + (-0.449 + 0.779i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + (7.83 + 13.5i)T + (-48.5 + 84.0i)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

−14.34682488027570077772998314399, −12.88587816488495172105977744887, −11.81817518642034313736296876105, −11.16023919221597904342252266492, −10.33226462264486833100974219959, −8.825428349477017618911683862160, −7.54794332249330086249902289740, −5.75126890898905878344211176071, −4.38592304884536978046225303563, −3.25585626399353990198167632791, 1.54151959001482983014460890972, 4.52032711300677936695871667532, 6.06677272383344985850022443475, 6.93921382332453102387667450538, 7.71857427659596176480219074764, 9.588854659042422940602621891710, 11.34290430013609946385398594773, 11.51088227671547309150968618632, 13.02596106157498602842496194692, 13.79657402858578961845453598371