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Properties

Label 2-693-7.4-c1-0-29
Degree $2$
Conductor $693$
Sign $-0.0725 - 0.997i$
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  + (−1.20 − 2.09i)2-s + (−1.91 + 3.31i)4-s + (−1 − 1.73i)5-s + (1 − 2.44i)7-s + 4.41·8-s + (−2.41 + 4.18i)10-s + (0.5 − 0.866i)11-s − 0.828·13-s + (−6.32 + 0.866i)14-s + (−1.49 − 2.59i)16-s + (−2.20 + 3.82i)17-s + (−3.62 − 6.27i)19-s + 7.65·20-s − 2.41·22-s + (−3.5 − 6.06i)23-s + ⋯
L(s)  = 1  + (−0.853 − 1.47i)2-s + (−0.957 + 1.65i)4-s + (−0.447 − 0.774i)5-s + (0.377 − 0.925i)7-s + 1.56·8-s + (−0.763 + 1.32i)10-s + (0.150 − 0.261i)11-s − 0.229·13-s + (−1.69 + 0.231i)14-s + (−0.374 − 0.649i)16-s + (−0.535 + 0.927i)17-s + (−0.830 − 1.43i)19-s + 1.71·20-s − 0.514·22-s + (−0.729 − 1.26i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.0725 - 0.997i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ -0.0725 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.249817 + 0.268653i\)
\(L(\frac12)\) \(\approx\) \(0.249817 + 0.268653i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.20 + 2.09i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 0.828T + 13T^{2} \)
17 \( 1 + (2.20 - 3.82i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.62 + 6.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.5 + 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.24T + 29T^{2} \)
31 \( 1 + (2.82 - 4.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.74 - 8.21i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.17T + 41T^{2} \)
43 \( 1 + 2.75T + 43T^{2} \)
47 \( 1 + (-4.91 - 8.51i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.58 - 6.21i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.32 - 7.49i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.58 + 2.74i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.17T + 71T^{2} \)
73 \( 1 + (0.171 - 0.297i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.65 + 11.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.82T + 83T^{2} \)
89 \( 1 + (7.07 + 12.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−10.03912840295362852762467993194, −8.910634418724824863149920962029, −8.526335005064436155372860056902, −7.65527998088111704573187057164, −6.41509778388546869023153831698, −4.55574107111046491697669877589, −4.17610305447655481376708177679, −2.78405705825801798717083721270, −1.46885159013585808002176385066, −0.26349781849665998357254841468, 2.11060010848395593634293077390, 3.82704953159946730164569158429, 5.25994514068242824263688028027, 5.94868598386468812987435310381, 6.90348092888623271254759601293, 7.64685115753745201717314152599, 8.248025338583741781792716148882, 9.279582472879225167105734179934, 9.778685863972598807884929699244, 10.96936133659775427078109945579

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