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Properties

Label 2-693-11.4-c1-0-13
Degree $2$
Conductor $693$
Sign $0.175 - 0.984i$
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  + (1.60 + 1.16i)2-s + (0.598 + 1.84i)4-s + (−0.0217 + 0.0158i)5-s + (0.309 + 0.951i)7-s + (0.0378 − 0.116i)8-s − 0.0534·10-s + (1.37 + 3.01i)11-s + (3.94 + 2.86i)13-s + (−0.613 + 1.88i)14-s + (3.33 − 2.42i)16-s + (1.35 − 0.986i)17-s + (0.424 − 1.30i)19-s + (−0.0422 − 0.0306i)20-s + (−1.31 + 6.45i)22-s − 8.06·23-s + ⋯
L(s)  = 1  + (1.13 + 0.824i)2-s + (0.299 + 0.921i)4-s + (−0.00974 + 0.00707i)5-s + (0.116 + 0.359i)7-s + (0.0133 − 0.0411i)8-s − 0.0168·10-s + (0.414 + 0.909i)11-s + (1.09 + 0.795i)13-s + (−0.163 + 0.504i)14-s + (0.833 − 0.605i)16-s + (0.329 − 0.239i)17-s + (0.0974 − 0.299i)19-s + (−0.00944 − 0.00685i)20-s + (−0.279 + 1.37i)22-s − 1.68·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.175 - 0.984i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ 0.175 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20485 + 1.84613i\)
\(L(\frac12)\) \(\approx\) \(2.20485 + 1.84613i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-1.37 - 3.01i)T \)
good2 \( 1 + (-1.60 - 1.16i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (0.0217 - 0.0158i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-3.94 - 2.86i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.35 + 0.986i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.424 + 1.30i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 8.06T + 23T^{2} \)
29 \( 1 + (-1.97 - 6.08i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.24 + 2.35i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.161 - 0.495i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.27 + 10.0i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.73T + 43T^{2} \)
47 \( 1 + (-2.76 + 8.52i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.20 + 2.32i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.00 + 9.25i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (6.85 - 4.97i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 2.81T + 67T^{2} \)
71 \( 1 + (-1.65 + 1.20i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.23 + 9.95i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.73 + 3.44i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.10 + 1.53i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 1.21T + 89T^{2} \)
97 \( 1 + (2.74 + 1.99i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.76254923024270678419448074783, −9.665536817921962867216099506076, −8.883251358329057026150420453257, −7.68763725706741599303068184979, −6.94221145499829348404330935291, −6.09219024853137283358358224397, −5.31507728375608356358225244077, −4.27493305574014417588725152405, −3.54849562347342657513193418210, −1.80662876067279089010283857602, 1.27167923188188725928020150029, 2.75854737808194058897202836827, 3.75613106920195665342967383759, 4.37638407002945784810434243028, 5.85123734611024177566216736658, 6.09445827765730054920003064239, 7.900128223744790036539430792697, 8.340378154380819807275012908187, 9.748628958805905708354808578226, 10.63498511247768760948075112524

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