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Properties

Label 2-693-11.3-c1-0-6
Degree $2$
Conductor $693$
Sign $0.530 - 0.847i$
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  + (−0.809 + 0.587i)2-s + (−0.309 + 0.951i)4-s + (−2.30 − 1.67i)5-s + (0.309 − 0.951i)7-s + (−0.927 − 2.85i)8-s + 2.85·10-s + (−3.04 + 1.31i)11-s + (1 − 0.726i)13-s + (0.309 + 0.951i)14-s + (0.809 + 0.587i)16-s + (6.35 + 4.61i)17-s + (0.809 + 2.48i)19-s + (2.30 − 1.67i)20-s + (1.69 − 2.85i)22-s + 3.09·23-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.154 + 0.475i)4-s + (−1.03 − 0.750i)5-s + (0.116 − 0.359i)7-s + (−0.327 − 1.00i)8-s + 0.902·10-s + (−0.918 + 0.396i)11-s + (0.277 − 0.201i)13-s + (0.0825 + 0.254i)14-s + (0.202 + 0.146i)16-s + (1.54 + 1.11i)17-s + (0.185 + 0.571i)19-s + (0.516 − 0.375i)20-s + (0.360 − 0.608i)22-s + 0.644·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.530 - 0.847i$
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: \((2,\ 693,\ (\ :1/2),\ 0.530 - 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.671152 + 0.371903i\)
\(L(\frac12)\) \(\approx\) \(0.671152 + 0.371903i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.04 - 1.31i)T \)
good2 \( 1 + (0.809 - 0.587i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (2.30 + 1.67i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-1 + 0.726i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-6.35 - 4.61i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.809 - 2.48i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.09T + 23T^{2} \)
29 \( 1 + (-0.618 + 1.90i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.92 + 4.30i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.73 - 11.4i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.35 - 10.3i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.23T + 43T^{2} \)
47 \( 1 + (0.618 + 1.90i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-9.85 + 7.15i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.09 - 6.43i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (7.23 + 5.25i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + (-7.23 - 5.25i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.61 - 4.97i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-11.3 + 8.22i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.38 - 1.73i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 1.09T + 89T^{2} \)
97 \( 1 + (-2.85 + 2.07i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.29727141295827589504629409606, −9.757936444172576688497310202019, −8.391442072393275178307995444086, −8.104268059346842530620547015105, −7.55900407625706994243786362772, −6.35979151157641625482737136225, −5.05180684247056546802936219739, −4.08411923501280390151760763337, −3.19632775311508471343966250268, −0.966344916145856446139442456855, 0.68925657900783039337643021203, 2.53534668409127850234191085406, 3.42180007586248160762830848745, 4.96850399786554731060142592050, 5.68881104414144643900580426039, 7.07492665886614031114258062952, 7.77917338123097078331045184954, 8.713048966992084143221275640597, 9.480685749568107493144463153249, 10.58523859964979087108464448386

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