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Properties

Label 2-693-7.2-c1-0-16
Degree $2$
Conductor $693$
Sign $0.777 - 0.629i$
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  + (−1.24 + 2.15i)2-s + (−2.10 − 3.64i)4-s + (1.10 − 1.90i)5-s + (−1.10 + 2.40i)7-s + 5.49·8-s + (2.74 + 4.75i)10-s + (−0.5 − 0.866i)11-s − 3.28·13-s + (−3.81 − 5.37i)14-s + (−2.63 + 4.56i)16-s + (0.745 + 1.29i)17-s + (3.45 − 5.99i)19-s − 9.26·20-s + 2.49·22-s + (3.24 − 5.62i)23-s + ⋯
L(s)  = 1  + (−0.880 + 1.52i)2-s + (−1.05 − 1.82i)4-s + (0.492 − 0.853i)5-s + (−0.416 + 0.909i)7-s + 1.94·8-s + (0.868 + 1.50i)10-s + (−0.150 − 0.261i)11-s − 0.911·13-s + (−1.01 − 1.43i)14-s + (−0.658 + 1.14i)16-s + (0.180 + 0.313i)17-s + (0.793 − 1.37i)19-s − 2.07·20-s + 0.531·22-s + (0.676 − 1.17i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 - 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.777 - 0.629i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ 0.777 - 0.629i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.782936 + 0.277340i\)
\(L(\frac12)\) \(\approx\) \(0.782936 + 0.277340i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.10 - 2.40i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (1.24 - 2.15i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.10 + 1.90i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 3.28T + 13T^{2} \)
17 \( 1 + (-0.745 - 1.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.45 + 5.99i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.24 + 5.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.64T + 29T^{2} \)
31 \( 1 + (-1.17 - 2.03i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.77 + 4.81i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 - 5.26T + 43T^{2} \)
47 \( 1 + (-0.745 + 1.29i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.152 - 0.263i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.32 + 10.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.49 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.28 - 3.96i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + (-4.28 - 7.41i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.31 - 4.01i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.93T + 83T^{2} \)
89 \( 1 + (-1.60 + 2.77i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.85T + 97T^{2} \)
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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

−10.00989839831852489788545950341, −9.197122834452520465466080470311, −8.981493392243101426106978318068, −8.004471208618553970517957428669, −7.07531559571689113432129196369, −6.21136922986886863373874681975, −5.35507625661096826105101103357, −4.78902684116010550395826980833, −2.61399679445366661060172809454, −0.71168542978710544177203465119, 1.15493543521656163124134729917, 2.54479652341114986569214699365, 3.29116946092428942354737408537, 4.39030731057562813620400017103, 5.94561698701805668182088435921, 7.32688738125986241241553386772, 7.72805165600239744306666904672, 9.167462757914893702166075958106, 9.875248514295824315161141890813, 10.21609913443666295791205980913

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