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Properties

Label 2-693-693.142-c0-0-1
Degree $2$
Conductor $693$
Sign $0.805 + 0.592i$
Analytic cond. $0.345852$
Root an. cond. $0.588091$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s − 3-s + (−0.866 + 0.5i)6-s − i·7-s + i·8-s + 9-s + 11-s + (0.866 − 0.5i)13-s + (−0.5 − 0.866i)14-s + (0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (0.866 − 0.5i)18-s + (−0.866 − 0.5i)19-s + i·21-s + (0.866 − 0.5i)22-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s − 3-s + (−0.866 + 0.5i)6-s − i·7-s + i·8-s + 9-s + 11-s + (0.866 − 0.5i)13-s + (−0.5 − 0.866i)14-s + (0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (0.866 − 0.5i)18-s + (−0.866 − 0.5i)19-s + i·21-s + (0.866 − 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.805 + 0.592i$
Analytic conductor: \(0.345852\)
Root analytic conductor: \(0.588091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :0),\ 0.805 + 0.592i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.109782521\)
\(L(\frac12)\) \(\approx\) \(1.109782521\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + iT \)
11 \( 1 - T \)
good2 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.81569738984036808023525311553, −10.15947854165825597136367123252, −8.930992622310572585628435029148, −7.83109156987871952573731382636, −6.83501692388892640027390601428, −5.95573859719963939034056248648, −4.93167957438484986995322351727, −4.12130707457387830698750161082, −3.32835334212576607343082347815, −1.40107305215020597884959158340, 1.60960709184498179182490488074, 3.73714501775804461804290552566, 4.42711058102940267492558436630, 5.65728463326764386465252228210, 6.05119525655453434985846127959, 6.70000569182616561835799847611, 7.997664096990617985039065147102, 9.195701347879954574245281395262, 9.894967219614745607330756441697, 10.92330477451665082125659386301

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