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Properties

Label 2-693-231.83-c0-0-2
Degree $2$
Conductor $693$
Sign $0.989 + 0.147i$
Analytic cond. $0.345852$
Root an. cond. $0.588091$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0966 + 0.297i)2-s + (0.729 − 0.530i)4-s + (−0.587 − 0.809i)7-s + (0.481 + 0.349i)8-s + (0.453 + 0.891i)11-s + (0.183 − 0.253i)14-s + (0.221 − 0.680i)16-s + (−0.221 + 0.221i)22-s − 1.97i·23-s + (0.809 + 0.587i)25-s + (−0.857 − 0.278i)28-s + (−1.44 + 1.04i)29-s + 0.819·32-s + (−0.951 + 0.690i)37-s + 0.618i·43-s + (0.803 + 0.409i)44-s + ⋯
L(s)  = 1  + (0.0966 + 0.297i)2-s + (0.729 − 0.530i)4-s + (−0.587 − 0.809i)7-s + (0.481 + 0.349i)8-s + (0.453 + 0.891i)11-s + (0.183 − 0.253i)14-s + (0.221 − 0.680i)16-s + (−0.221 + 0.221i)22-s − 1.97i·23-s + (0.809 + 0.587i)25-s + (−0.857 − 0.278i)28-s + (−1.44 + 1.04i)29-s + 0.819·32-s + (−0.951 + 0.690i)37-s + 0.618i·43-s + (0.803 + 0.409i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.989 + 0.147i$
Analytic conductor: \(0.345852\)
Root analytic conductor: \(0.588091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (314, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :0),\ 0.989 + 0.147i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.135531082\)
\(L(\frac12)\) \(\approx\) \(1.135531082\)
\(L(1)\) 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 \( 1 + (0.587 + 0.809i)T \)
11 \( 1 + (-0.453 - 0.891i)T \)
good2 \( 1 + (-0.0966 - 0.297i)T + (-0.809 + 0.587i)T^{2} \)
5 \( 1 + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T^{2} \)
23 \( 1 + 1.97iT - T^{2} \)
29 \( 1 + (1.44 - 1.04i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - 0.618iT - T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (1.69 - 0.550i)T + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.809 - 0.587i)T^{2} \)
67 \( 1 - 0.618T + T^{2} \)
71 \( 1 + (-0.863 - 0.280i)T + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−10.62603049414089980484833449732, −9.934857707431957513819544215662, −9.044998901137301368863374005594, −7.78893191747095180800801862526, −6.85882699432393776811435476607, −6.56697787604380573756304544863, −5.26252538774297354211240928602, −4.29780270808106700336652413382, −2.96516354527302468125286854457, −1.53608569331353593124517473024, 1.89201622697530961267569647771, 3.11916245426101251839014260476, 3.81393599566229795282821186857, 5.45695058074476402527616468711, 6.23857367530578896398124105634, 7.16195617016528830367786657062, 8.101487919443979394264245239725, 9.027147898337136546181367259995, 9.805715539871308135315063452952, 10.96035597988833583256684045708

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