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Properties

Label 2-693-7.4-c1-0-0
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 $0$

Origins

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Normalization:  

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: $\chi_{693} (298, \cdot )$
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

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

Graph of the $Z$-function along the critical line