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Properties

Label 2-693-77.69-c0-0-1
Degree $2$
Conductor $693$
Sign $0.999 + 0.0237i$
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  + (0.5 − 0.363i)2-s + (−0.190 + 0.587i)4-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (0.809 + 0.587i)11-s + (−0.190 − 0.587i)14-s + 0.618·22-s − 0.618·23-s + (0.309 + 0.951i)25-s + (0.5 + 0.363i)28-s + (0.5 − 1.53i)29-s − 32-s + (0.190 − 0.587i)37-s − 1.61·43-s + (−0.5 + 0.363i)44-s + ⋯
L(s)  = 1  + (0.5 − 0.363i)2-s + (−0.190 + 0.587i)4-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (0.809 + 0.587i)11-s + (−0.190 − 0.587i)14-s + 0.618·22-s − 0.618·23-s + (0.309 + 0.951i)25-s + (0.5 + 0.363i)28-s + (0.5 − 1.53i)29-s − 32-s + (0.190 − 0.587i)37-s − 1.61·43-s + (−0.5 + 0.363i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.999 + 0.0237i$
Analytic conductor: \(0.345852\)
Root analytic conductor: \(0.588091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :0),\ 0.999 + 0.0237i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.217269030\)
\(L(\frac12)\) \(\approx\) \(1.217269030\)
\(L(1)\) 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 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
5 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + 0.618T + T^{2} \)
29 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 + 1.61T + T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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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

−10.83301681733723041346287853852, −9.893517205601291695119718822597, −8.968929824521553842450707112374, −7.925261414363894252373646350795, −7.32474443001207576625097753417, −6.21909408234073838145710325050, −4.81523183367441219737895470928, −4.18889246149609497396793835941, −3.26139941008477391550068681989, −1.78605585050675089750236965703, 1.57042847074569540143355802166, 3.19473303539169765575050987320, 4.46240226692688232180939540174, 5.28591457278548246928516887718, 6.18390267531503340560092388832, 6.79147399468227445010776211233, 8.218766942795135459760244403772, 8.909348253926549395740526823902, 9.779916010640865863200040405013, 10.65062822668939876413655651983

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