Properties

 Label 2-693-7.2-c1-0-19 Degree $2$ Conductor $693$ Sign $0.550 + 0.834i$ Analytic cond. $5.53363$ Root an. cond. $2.35236$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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Dirichlet series

 L(s)  = 1 + (−0.758 + 1.31i)2-s + (−0.150 − 0.259i)4-s + (−1.16 + 2.02i)5-s + (−2.28 − 1.33i)7-s − 2.57·8-s + (−1.76 − 3.06i)10-s + (−0.5 − 0.866i)11-s − 1.53·13-s + (3.48 − 1.99i)14-s + (2.25 − 3.90i)16-s + (−0.0583 − 0.100i)17-s + (1.80 − 3.12i)19-s + 0.699·20-s + 1.51·22-s + (3.55 − 6.15i)23-s + ⋯
 L(s)  = 1 + (−0.536 + 0.928i)2-s + (−0.0750 − 0.129i)4-s + (−0.521 + 0.903i)5-s + (−0.863 − 0.503i)7-s − 0.911·8-s + (−0.559 − 0.969i)10-s + (−0.150 − 0.261i)11-s − 0.426·13-s + (0.930 − 0.532i)14-s + (0.563 − 0.976i)16-s + (−0.0141 − 0.0244i)17-s + (0.414 − 0.717i)19-s + 0.156·20-s + 0.323·22-s + (0.741 − 1.28i)23-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 + 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.550 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$693$$    =    $$3^{2} \cdot 7 \cdot 11$$ Sign: $0.550 + 0.834i$ Analytic conductor: $$5.53363$$ Root analytic conductor: $$2.35236$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{693} (100, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 693,\ (\ :1/2),\ 0.550 + 0.834i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.186361 - 0.100357i$$ $$L(\frac12)$$ $$\approx$$ $$0.186361 - 0.100357i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1 + (2.28 + 1.33i)T$$
11 $$1 + (0.5 + 0.866i)T$$
good2 $$1 + (0.758 - 1.31i)T + (-1 - 1.73i)T^{2}$$
5 $$1 + (1.16 - 2.02i)T + (-2.5 - 4.33i)T^{2}$$
13 $$1 + 1.53T + 13T^{2}$$
17 $$1 + (0.0583 + 0.100i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (-1.80 + 3.12i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (-3.55 + 6.15i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + 5.05T + 29T^{2}$$
31 $$1 + (-2.18 - 3.79i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (-0.150 + 0.259i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + 8.20T + 41T^{2}$$
43 $$1 + 4.18T + 43T^{2}$$
47 $$1 + (-1.15 + 1.99i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (5.94 + 10.2i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (-2.47 - 4.29i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-2.28 + 3.95i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (6.14 + 10.6i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + 4.25T + 71T^{2}$$
73 $$1 + (5.21 + 9.03i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (7.15 - 12.3i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 - 11.1T + 83T^{2}$$
89 $$1 + (-6.96 + 12.0i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + 16.4T + 97T^{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

−10.22182805205087527535624239134, −9.316731177253385774179735144737, −8.460053128165468162398338726376, −7.48026475824604482800141116044, −6.90243085743208671206354804041, −6.38582572749137042597449981868, −5.04441954478427094879258538101, −3.50979432976736080367461554765, −2.83902839336353588532974381923, −0.13425288662987233237221666298, 1.41101548088381238794483599754, 2.77314863342186153824634007632, 3.78819041028867657200344784851, 5.16350553492009328718268600926, 6.01875493226467586437224814833, 7.26044915931611588173062965261, 8.312035938712336504563642864025, 9.182172064503575563342186890494, 9.674656879469007456374225850571, 10.46836654489578786748565337818