↑

Properties

Label 2-693-11.5-c1-0-12
Degree $2$
Conductor $693$
Sign $-0.371 + 0.928i$
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  + (−0.864 − 2.66i)2-s + (−4.71 + 3.42i)4-s + (0.518 − 1.59i)5-s + (0.809 − 0.587i)7-s + (8.67 + 6.30i)8-s − 4.69·10-s + (3.13 + 1.09i)11-s + (1.81 + 5.58i)13-s + (−2.26 − 1.64i)14-s + (5.67 − 17.4i)16-s + (0.0939 − 0.289i)17-s + (3.74 + 2.72i)19-s + (3.02 + 9.30i)20-s + (0.194 − 9.28i)22-s + 1.38·23-s + ⋯
L(s)  = 1  + (−0.611 − 1.88i)2-s + (−2.35 + 1.71i)4-s + (0.231 − 0.713i)5-s + (0.305 − 0.222i)7-s + (3.06 + 2.22i)8-s − 1.48·10-s + (0.944 + 0.328i)11-s + (0.503 + 1.54i)13-s + (−0.605 − 0.439i)14-s + (1.41 − 4.36i)16-s + (0.0227 − 0.0701i)17-s + (0.859 + 0.624i)19-s + (0.676 + 2.08i)20-s + (0.0414 − 1.97i)22-s + 0.288·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.371 + 0.928i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ -0.371 + 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.624930 - 0.923391i\)
\(L(\frac12)\) \(\approx\) \(0.624930 - 0.923391i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-3.13 - 1.09i)T \)
good2 \( 1 + (0.864 + 2.66i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-0.518 + 1.59i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (-1.81 - 5.58i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.0939 + 0.289i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.74 - 2.72i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 1.38T + 23T^{2} \)
29 \( 1 + (-3.21 + 2.33i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.236 - 0.726i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.59 - 1.88i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.72 + 4.88i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.01T + 43T^{2} \)
47 \( 1 + (2.09 + 1.52i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.449 - 1.38i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.01 - 1.46i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.04 + 3.22i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 6.16T + 67T^{2} \)
71 \( 1 + (2.34 - 7.22i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-10.3 + 7.49i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.90 + 5.85i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.621 + 1.91i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 0.0843T + 89T^{2} \)
97 \( 1 + (-0.320 - 0.986i)T + (-78.4 + 57.0i)T^{2} \)
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

−10.18290451518205834479761528495, −9.381462371979823381225285728223, −8.934699530383960875419482154902, −8.124618637472008510474586732055, −6.89986570030661644843069720145, −5.10775662011914847947638494479, −4.28560748748375669727882815617, −3.45778641167133540563574726488, −1.89230621170908087854535577187, −1.17797559751890108737837791313, 1.00567519624969199959174381966, 3.38564227599195628499743693314, 4.84421143598895280260558679592, 5.65099968303358526134432959656, 6.47267418312385174651125838382, 7.12269951328649335562598679626, 8.118100887482236752010637926362, 8.684249747336961124961581344668, 9.619104598670811760444565890219, 10.37085549411436244084815270880

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