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Properties

Label 2-693-11.3-c1-0-22
Degree $2$
Conductor $693$
Sign $0.751 + 0.659i$
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  + (−1.80 + 1.31i)2-s + (0.927 − 2.85i)4-s + (1.30 + 0.951i)5-s + (0.309 − 0.951i)7-s + (0.690 + 2.12i)8-s − 3.61·10-s + (−0.809 − 3.21i)11-s + (1 − 0.726i)13-s + (0.690 + 2.12i)14-s + (0.809 + 0.587i)16-s + (−1.5 − 1.08i)17-s + (−1.19 − 3.66i)19-s + (3.92 − 2.85i)20-s + (5.69 + 4.75i)22-s − 6.61·23-s + ⋯
L(s)  = 1  + (−1.27 + 0.929i)2-s + (0.463 − 1.42i)4-s + (0.585 + 0.425i)5-s + (0.116 − 0.359i)7-s + (0.244 + 0.751i)8-s − 1.14·10-s + (−0.243 − 0.969i)11-s + (0.277 − 0.201i)13-s + (0.184 + 0.568i)14-s + (0.202 + 0.146i)16-s + (−0.363 − 0.264i)17-s + (−0.273 − 0.840i)19-s + (0.878 − 0.637i)20-s + (1.21 + 1.01i)22-s − 1.37·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.751 + 0.659i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ 0.751 + 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.550199 - 0.207045i\)
\(L(\frac12)\) \(\approx\) \(0.550199 - 0.207045i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 + 3.21i)T \)
good2 \( 1 + (1.80 - 1.31i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-1.30 - 0.951i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-1 + 0.726i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.5 + 1.08i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.19 + 3.66i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 6.61T + 23T^{2} \)
29 \( 1 + (-1.85 + 5.70i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (6.54 - 4.75i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.736 + 2.26i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.97 + 9.14i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.70T + 43T^{2} \)
47 \( 1 + (-1.38 - 4.25i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-9.09 + 6.60i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.381 + 1.17i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + (-10.4 - 7.60i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.0901 + 0.277i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-8.09 + 5.87i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-6.09 - 4.42i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 3.38T + 89T^{2} \)
97 \( 1 + (-4.85 + 3.52i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.28663746334746401883482577079, −9.338866733635154067186156080633, −8.568725531926419280914353870452, −7.87414515323457116536314192254, −6.90174671999689908626027482301, −6.23066863500729309798997485019, −5.39207939116513542132151188738, −3.76784620495717196351042213406, −2.18883715489788826614869612617, −0.46585796852888750620517922403, 1.57314477923992128282066182066, 2.23324114731280969912239438286, 3.73476078243174756628525878601, 5.10506057260647229504080679984, 6.16868064073220007114078794748, 7.46806371381200743035634387499, 8.288000660083476111268661260961, 9.036253646577604834494295122452, 9.773876208856025714074923380303, 10.30570773385873398455901927309

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