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Properties

Label 2-693-11.9-c1-0-27
Degree $2$
Conductor $693$
Sign $-0.968 - 0.247i$
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  + (0.648 − 1.99i)2-s + (−1.94 − 1.41i)4-s + (−0.976 − 3.00i)5-s + (0.809 + 0.587i)7-s + (−0.679 + 0.493i)8-s − 6.62·10-s + (0.965 − 3.17i)11-s + (0.657 − 2.02i)13-s + (1.69 − 1.23i)14-s + (−0.939 − 2.89i)16-s + (1.48 + 4.55i)17-s + (−1.24 + 0.904i)19-s + (−2.34 + 7.21i)20-s + (−5.70 − 3.98i)22-s − 5.11·23-s + ⋯
L(s)  = 1  + (0.458 − 1.41i)2-s + (−0.970 − 0.705i)4-s + (−0.436 − 1.34i)5-s + (0.305 + 0.222i)7-s + (−0.240 + 0.174i)8-s − 2.09·10-s + (0.291 − 0.956i)11-s + (0.182 − 0.561i)13-s + (0.453 − 0.329i)14-s + (−0.234 − 0.722i)16-s + (0.359 + 1.10i)17-s + (−0.285 + 0.207i)19-s + (−0.523 + 1.61i)20-s + (−1.21 − 0.849i)22-s − 1.06·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.968 - 0.247i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ -0.968 - 0.247i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.210539 + 1.67702i\)
\(L(\frac12)\) \(\approx\) \(0.210539 + 1.67702i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.965 + 3.17i)T \)
good2 \( 1 + (-0.648 + 1.99i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (0.976 + 3.00i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-0.657 + 2.02i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.48 - 4.55i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.24 - 0.904i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 5.11T + 23T^{2} \)
29 \( 1 + (0.775 + 0.563i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.23 - 6.86i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.94 + 1.41i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.215 + 0.156i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 9.28T + 43T^{2} \)
47 \( 1 + (-8.35 + 6.06i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.292 + 0.899i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (7.72 + 5.61i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.64 - 8.13i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 13.2T + 67T^{2} \)
71 \( 1 + (4.37 + 13.4i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.24 + 4.53i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.87 + 8.85i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-5.20 - 16.0i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 + (-2.19 + 6.74i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.36624064144327565810441692146, −9.157693869357468674387728845325, −8.534388866760589012998163230381, −7.71824117494877523059027825608, −5.99175463274348393705646622446, −5.14926458367824329491805820928, −4.11958429174221237730660742041, −3.45634787533978788282205485183, −1.91220342882383478664557797516, −0.801164942532071048920173128654, 2.30647601464560909229162406639, 3.84846703311909053074629179217, 4.58159266518746687924645624954, 5.81597791355452969776001012846, 6.64291048838440678129246145240, 7.39333615172761527534703261451, 7.69873627823125244418190214606, 9.014028714185221878843590592232, 10.01746767434267089383008260795, 11.00519355590568342142303536669

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