# Properties

 Label 2-91-91.4-c1-0-6 Degree $2$ Conductor $91$ Sign $-0.968 + 0.250i$ 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 − 2.58i·2-s + (−0.259 − 0.449i)3-s − 4.70·4-s + (−1.39 + 0.806i)5-s + (−1.16 + 0.671i)6-s + (1.06 − 2.42i)7-s + 6.99i·8-s + (1.36 − 2.36i)9-s + (2.08 + 3.61i)10-s + (2.34 − 1.35i)11-s + (1.21 + 2.11i)12-s + (2.36 + 2.71i)13-s + (−6.27 − 2.74i)14-s + (0.723 + 0.417i)15-s + 8.69·16-s − 3.12·17-s + ⋯
 L(s)  = 1 − 1.83i·2-s + (−0.149 − 0.259i)3-s − 2.35·4-s + (−0.624 + 0.360i)5-s + (−0.474 + 0.273i)6-s + (0.401 − 0.915i)7-s + 2.47i·8-s + (0.455 − 0.788i)9-s + (0.659 + 1.14i)10-s + (0.706 − 0.407i)11-s + (0.351 + 0.609i)12-s + (0.656 + 0.753i)13-s + (−1.67 − 0.734i)14-s + (0.186 + 0.107i)15-s + 2.17·16-s − 0.758·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$91$$    =    $$7 \cdot 13$$ Sign: $-0.968 + 0.250i$ Analytic conductor: $$0.726638$$ Root analytic conductor: $$0.852431$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{91} (4, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 91,\ (\ :1/2),\ -0.968 + 0.250i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.102881 - 0.808077i$$ $$L(\frac12)$$ $$\approx$$ $$0.102881 - 0.808077i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1 + (-1.06 + 2.42i)T$$
13 $$1 + (-2.36 - 2.71i)T$$
good2 $$1 + 2.58iT - 2T^{2}$$
3 $$1 + (0.259 + 0.449i)T + (-1.5 + 2.59i)T^{2}$$
5 $$1 + (1.39 - 0.806i)T + (2.5 - 4.33i)T^{2}$$
11 $$1 + (-2.34 + 1.35i)T + (5.5 - 9.52i)T^{2}$$
17 $$1 + 3.12T + 17T^{2}$$
19 $$1 + (-3.18 - 1.84i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 + 1.98T + 23T^{2}$$
29 $$1 + (-2.68 + 4.65i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (-9.07 - 5.23i)T + (15.5 + 26.8i)T^{2}$$
37 $$1 - 5.95iT - 37T^{2}$$
41 $$1 + (6.66 + 3.85i)T + (20.5 + 35.5i)T^{2}$$
43 $$1 + (1.67 + 2.90i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (-0.913 + 0.527i)T + (23.5 - 40.7i)T^{2}$$
53 $$1 + (3.63 - 6.29i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 - 11.4iT - 59T^{2}$$
61 $$1 + (-1.46 + 2.53i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-11.7 + 6.79i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 + (-1.17 + 0.675i)T + (35.5 - 61.4i)T^{2}$$
73 $$1 + (7.88 + 4.55i)T + (36.5 + 63.2i)T^{2}$$
79 $$1 + (-3.10 - 5.37i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 2.69iT - 83T^{2}$$
89 $$1 - 1.75iT - 89T^{2}$$
97 $$1 + (13.4 - 7.74i)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

−13.53429236003749609406744102316, −11.98949995808972793448462448610, −11.66814975298971300007850876850, −10.63864451778223732063958098317, −9.603173921960597174735709023613, −8.360790708578977239248235477625, −6.71485993270863562220455265898, −4.33935097469234951997144780227, −3.52586268377497243789815218450, −1.26612256890437437309622037185, 4.34501875962581247912489375981, 5.27188710414152442926613317590, 6.55829517220568463113799243894, 7.908185700752904829056834577693, 8.547437872865687811885774012356, 9.768032453162149619297677914334, 11.49575939854236929926360378239, 12.82081517807083239645848257628, 13.86199443567006061363207153034, 14.97118413477645884851991268147