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Properties

Label 2-693-11.4-c1-0-25
Degree $2$
Conductor $693$
Sign $0.851 - 0.524i$
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  + (1.70 + 1.23i)2-s + (0.749 + 2.30i)4-s + (3.03 − 2.20i)5-s + (0.309 + 0.951i)7-s + (−0.277 + 0.853i)8-s + 7.89·10-s + (1.33 − 3.03i)11-s + (−2.80 − 2.03i)13-s + (−0.650 + 2.00i)14-s + (2.39 − 1.74i)16-s + (−1.23 + 0.897i)17-s + (−1.59 + 4.90i)19-s + (7.36 + 5.34i)20-s + (6.02 − 3.51i)22-s − 4.87·23-s + ⋯
L(s)  = 1  + (1.20 + 0.874i)2-s + (0.374 + 1.15i)4-s + (1.35 − 0.985i)5-s + (0.116 + 0.359i)7-s + (−0.0981 + 0.301i)8-s + 2.49·10-s + (0.402 − 0.915i)11-s + (−0.777 − 0.564i)13-s + (−0.173 + 0.534i)14-s + (0.599 − 0.435i)16-s + (−0.299 + 0.217i)17-s + (−0.365 + 1.12i)19-s + (1.64 + 1.19i)20-s + (1.28 − 0.749i)22-s − 1.01·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.851 - 0.524i$
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: \((2,\ 693,\ (\ :1/2),\ 0.851 - 0.524i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.31354 + 0.937945i\)
\(L(\frac12)\) \(\approx\) \(3.31354 + 0.937945i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.33 + 3.03i)T \)
good2 \( 1 + (-1.70 - 1.23i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-3.03 + 2.20i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (2.80 + 2.03i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.23 - 0.897i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.59 - 4.90i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 4.87T + 23T^{2} \)
29 \( 1 + (-3.31 - 10.2i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.57 + 4.77i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.86 - 8.80i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.36 - 10.3i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 0.137T + 43T^{2} \)
47 \( 1 + (0.843 - 2.59i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.26 - 1.64i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.49 + 4.60i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.240 + 0.175i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 0.816T + 67T^{2} \)
71 \( 1 + (4.63 - 3.36i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.764 + 2.35i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (9.24 + 6.71i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-8.61 + 6.25i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 8.91T + 89T^{2} \)
97 \( 1 + (6.54 + 4.75i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.36497174062175256630261685176, −9.642265949738932983337702832930, −8.653687857940626128498952080345, −7.87695466732780230950624909167, −6.48638368026065033155855070134, −5.91830227579502395456785226489, −5.27927494972163344463787285886, −4.46172620468202673067770682233, −3.13551855364081359714239319083, −1.57657877016684890683581071460, 2.08527405408495588866683299656, 2.36994516527466007832149354359, 3.83853772108581985021469578048, 4.72421050053297060456140524958, 5.71094620717153595170654574509, 6.63780405164881173013024320095, 7.37749541600976664504269560966, 9.064954832361432919644699845572, 9.930038511666301745536565850649, 10.47483824722133396508055238726

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