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Properties

Label 2-693-11.9-c1-0-15
Degree $2$
Conductor $693$
Sign $0.0219 - 0.999i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0501 + 0.154i)2-s + (1.59 + 1.16i)4-s + (1.35 + 4.17i)5-s + (0.809 + 0.587i)7-s + (−0.521 + 0.378i)8-s − 0.711·10-s + (3.30 + 0.224i)11-s + (0.517 − 1.59i)13-s + (−0.131 + 0.0953i)14-s + (1.18 + 3.65i)16-s + (−1.91 − 5.88i)17-s + (2.81 − 2.04i)19-s + (−2.67 + 8.23i)20-s + (−0.200 + 0.499i)22-s − 0.568·23-s + ⋯
L(s)  = 1  + (−0.0354 + 0.109i)2-s + (0.798 + 0.580i)4-s + (0.606 + 1.86i)5-s + (0.305 + 0.222i)7-s + (−0.184 + 0.133i)8-s − 0.224·10-s + (0.997 + 0.0676i)11-s + (0.143 − 0.442i)13-s + (−0.0350 + 0.0254i)14-s + (0.296 + 0.913i)16-s + (−0.463 − 1.42i)17-s + (0.645 − 0.468i)19-s + (−0.598 + 1.84i)20-s + (−0.0427 + 0.106i)22-s − 0.118·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.0219 - 0.999i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ 0.0219 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49173 + 1.45929i\)
\(L(\frac12)\) \(\approx\) \(1.49173 + 1.45929i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.30 - 0.224i)T \)
good2 \( 1 + (0.0501 - 0.154i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-1.35 - 4.17i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-0.517 + 1.59i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.91 + 5.88i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.81 + 2.04i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 0.568T + 23T^{2} \)
29 \( 1 + (7.17 + 5.21i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.33 + 4.12i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.784 + 0.569i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.67 + 3.39i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 5.04T + 43T^{2} \)
47 \( 1 + (-3.78 + 2.74i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.735 + 2.26i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.30 + 1.67i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.87 - 8.84i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 7.14T + 67T^{2} \)
71 \( 1 + (-0.245 - 0.755i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.93 + 1.40i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.207 - 0.637i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.86 - 8.83i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 + (-4.65 + 14.3i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.86246828720029661417567254171, −9.873277546189073413437204740545, −9.086906182438114279183896409873, −7.65962155292146516700251334967, −7.17200227966339886063711108018, −6.41916222874017338386972854924, −5.62354333512271857433834996239, −3.84886706384851472838047957155, −2.86893548354885915628401937245, −2.11864455355143724306131944009, 1.29117140119591295697529726053, 1.77808332283637174412816484617, 3.81716348518692776405403369636, 4.86350769255166729913368608389, 5.76943009928948058649011297471, 6.46324635134012591704662167885, 7.73663282692548122017660754097, 8.770279852625283213381854767594, 9.304461298986549927804801457253, 10.17922529519595810801561716600

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