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Properties

Label 2-693-77.20-c0-0-0
Degree $2$
Conductor $693$
Sign $-0.352 - 0.935i$
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  + (−0.363 + 1.11i)2-s + (−0.309 − 0.224i)4-s + (0.809 + 0.587i)7-s + (−0.587 + 0.427i)8-s + (0.951 − 0.309i)11-s + (−0.951 + 0.690i)14-s + (−0.381 − 1.17i)16-s + 1.17i·22-s − 1.17·23-s + (−0.809 + 0.587i)25-s + (−0.118 − 0.363i)28-s + (1.53 + 1.11i)29-s + 0.726·32-s + (−1.30 − 0.951i)37-s − 0.618·43-s + (−0.363 − 0.118i)44-s + ⋯
L(s)  = 1  + (−0.363 + 1.11i)2-s + (−0.309 − 0.224i)4-s + (0.809 + 0.587i)7-s + (−0.587 + 0.427i)8-s + (0.951 − 0.309i)11-s + (−0.951 + 0.690i)14-s + (−0.381 − 1.17i)16-s + 1.17i·22-s − 1.17·23-s + (−0.809 + 0.587i)25-s + (−0.118 − 0.363i)28-s + (1.53 + 1.11i)29-s + 0.726·32-s + (−1.30 − 0.951i)37-s − 0.618·43-s + (−0.363 − 0.118i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.352 - 0.935i$
Analytic conductor: \(0.345852\)
Root analytic conductor: \(0.588091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :0),\ -0.352 - 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8778344969\)
\(L(\frac12)\) \(\approx\) \(0.8778344969\)
\(L(1)\) 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 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.951 + 0.309i)T \)
good2 \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 + 1.17T + T^{2} \)
29 \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + 0.618T + T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 - 0.618T + T^{2} \)
71 \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
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

−10.97758878466157446955575407010, −9.800878084237018073496924516987, −8.799549341304837911891959783568, −8.401773541366934433709462861111, −7.45898845806986108300296822785, −6.57973129104627208242070417801, −5.77676202421178713006679659821, −4.89887025060322537751433606281, −3.48915664241004686197831111035, −1.93880397714047038183502271154, 1.27206220601000518827606570920, 2.34249843542874706414799064936, 3.76024141342393917354814362520, 4.51672299429528885848251292800, 6.01736629992327894967062892086, 6.89598351677328568201747165889, 8.067015976430216746431074467422, 8.822033513655187261462923709755, 10.07198391705505832211380474643, 10.15378068704614636933467790192

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