↑

Properties

Label 2-693-11.3-c1-0-20
Degree $2$
Conductor $693$
Sign $0.0326 + 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.07 + 0.783i)2-s + (−0.0686 + 0.211i)4-s + (−0.706 − 0.513i)5-s + (−0.309 + 0.951i)7-s + (−0.915 − 2.81i)8-s + 1.16·10-s + (−1.79 + 2.78i)11-s + (−0.0319 + 0.0232i)13-s + (−0.412 − 1.26i)14-s + (2.83 + 2.06i)16-s + (1.65 + 1.20i)17-s + (−1.75 − 5.41i)19-s + (0.157 − 0.114i)20-s + (−0.249 − 4.41i)22-s − 5.97·23-s + ⋯
L(s)  = 1  + (−0.762 + 0.554i)2-s + (−0.0343 + 0.105i)4-s + (−0.315 − 0.229i)5-s + (−0.116 + 0.359i)7-s + (−0.323 − 0.996i)8-s + 0.368·10-s + (−0.541 + 0.840i)11-s + (−0.00887 + 0.00644i)13-s + (−0.110 − 0.338i)14-s + (0.709 + 0.515i)16-s + (0.401 + 0.291i)17-s + (−0.403 − 1.24i)19-s + (0.0351 − 0.0255i)20-s + (−0.0531 − 0.941i)22-s − 1.24·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0326 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0326 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.0326 + 0.999i$
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: \((2,\ 693,\ (\ :1/2),\ 0.0326 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.171594 - 0.166087i\)
\(L(\frac12)\) \(\approx\) \(0.171594 - 0.166087i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.79 - 2.78i)T \)
good2 \( 1 + (1.07 - 0.783i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (0.706 + 0.513i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (0.0319 - 0.0232i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.65 - 1.20i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.75 + 5.41i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 5.97T + 23T^{2} \)
29 \( 1 + (-1.44 + 4.44i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (6.05 - 4.39i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.57 + 11.0i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.90 - 5.84i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.79T + 43T^{2} \)
47 \( 1 + (1.86 + 5.75i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.100 - 0.0731i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-4.43 + 13.6i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (6.11 + 4.44i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 4.85T + 67T^{2} \)
71 \( 1 + (9.10 + 6.61i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.189 + 0.582i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-0.125 + 0.0912i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.21 - 5.24i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 5.17T + 89T^{2} \)
97 \( 1 + (6.42 - 4.66i)T + (29.9 - 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.926977659290651000104458691514, −9.333583189854788053585528456114, −8.373271117320234933851289546865, −7.81026301057717277972755322162, −6.94716378042157732954794441900, −6.01216451379157206944131542794, −4.74278960572104795676512410261, −3.73401596111473271374615296540, −2.27671351149355339403452019309, −0.16508589987814056756993971245, 1.43395264490970736594859127286, 2.85409541218218954018407108131, 3.96381435099460451431220786411, 5.39938091442243511498652803407, 6.12591917310596058186226065070, 7.54044711833995843757176425946, 8.163111259390630925294090576092, 9.051884793426738686866141653528, 10.03693654473741483727550400714, 10.47757912273338999128105107312

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