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Properties

Label 2-693-7.2-c1-0-14
Degree $2$
Conductor $693$
Sign $0.749 - 0.661i$
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  + (0.207 − 0.358i)2-s + (0.914 + 1.58i)4-s + (−1 + 1.73i)5-s + (1 − 2.44i)7-s + 1.58·8-s + (0.414 + 0.717i)10-s + (0.5 + 0.866i)11-s + 4.82·13-s + (−0.671 − 0.866i)14-s + (−1.49 + 2.59i)16-s + (−0.792 − 1.37i)17-s + (0.621 − 1.07i)19-s − 3.65·20-s + 0.414·22-s + (−3.5 + 6.06i)23-s + ⋯
L(s)  = 1  + (0.146 − 0.253i)2-s + (0.457 + 0.791i)4-s + (−0.447 + 0.774i)5-s + (0.377 − 0.925i)7-s + 0.560·8-s + (0.130 + 0.226i)10-s + (0.150 + 0.261i)11-s + 1.33·13-s + (−0.179 − 0.231i)14-s + (−0.374 + 0.649i)16-s + (−0.192 − 0.333i)17-s + (0.142 − 0.246i)19-s − 0.817·20-s + 0.0883·22-s + (−0.729 + 1.26i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.749 - 0.661i$
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: \((2,\ 693,\ (\ :1/2),\ 0.749 - 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73030 + 0.654233i\)
\(L(\frac12)\) \(\approx\) \(1.73030 + 0.654233i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 2.44i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.207 + 0.358i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 4.82T + 13T^{2} \)
17 \( 1 + (0.792 + 1.37i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.621 + 1.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.5 - 6.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.24T + 29T^{2} \)
31 \( 1 + (-2.82 - 4.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.74 - 6.48i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.82T + 41T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
47 \( 1 + (-2.08 + 3.61i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.41 + 11.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.32 - 2.30i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.41 - 7.64i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.82T + 71T^{2} \)
73 \( 1 + (5.82 + 10.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.65 + 8.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.82T + 83T^{2} \)
89 \( 1 + (-7.07 + 12.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.48T + 97T^{2} \)
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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

−10.75061543721453703159624534156, −10.00314366170677102865897696512, −8.582570558830357237326798258709, −7.890146093081552971220344248029, −7.03965608739081474603488662510, −6.48378498101245259094655588796, −4.83662481099467662500769587868, −3.73115674347730912919928512399, −3.19269249959945522914473537213, −1.58668567852865526633602000635, 1.06995124577920140136894328316, 2.38695914952170782541225212293, 4.05984568664831566610473465616, 4.98677267985564683404868971079, 5.97743192435922022690245160041, 6.48737470772566669022260951376, 8.003884511331757266236350164110, 8.499599837811631794113213557734, 9.384983868539522889868101895338, 10.51807151679218105731244380621

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