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Properties

Label 2-693-231.41-c0-0-0
Degree $2$
Conductor $693$
Sign $0.402 - 0.915i$
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  + (−0.734 + 0.533i)2-s + (−0.0542 + 0.166i)4-s + (0.951 + 0.309i)7-s + (−0.329 − 1.01i)8-s + (0.987 − 0.156i)11-s + (−0.863 + 0.280i)14-s + (0.642 + 0.466i)16-s + (−0.642 + 0.642i)22-s + 1.78i·23-s + (−0.309 − 0.951i)25-s + (−0.103 + 0.142i)28-s + (−0.0966 + 0.297i)29-s + 0.346·32-s + (−0.587 + 1.80i)37-s − 1.61i·43-s + (−0.0274 + 0.173i)44-s + ⋯
L(s)  = 1  + (−0.734 + 0.533i)2-s + (−0.0542 + 0.166i)4-s + (0.951 + 0.309i)7-s + (−0.329 − 1.01i)8-s + (0.987 − 0.156i)11-s + (−0.863 + 0.280i)14-s + (0.642 + 0.466i)16-s + (−0.642 + 0.642i)22-s + 1.78i·23-s + (−0.309 − 0.951i)25-s + (−0.103 + 0.142i)28-s + (−0.0966 + 0.297i)29-s + 0.346·32-s + (−0.587 + 1.80i)37-s − 1.61i·43-s + (−0.0274 + 0.173i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.402 - 0.915i$
Analytic conductor: \(0.345852\)
Root analytic conductor: \(0.588091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :0),\ 0.402 - 0.915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7061868735\)
\(L(\frac12)\) \(\approx\) \(0.7061868735\)
\(L(1)\) 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 \( 1 + (-0.951 - 0.309i)T \)
11 \( 1 + (-0.987 + 0.156i)T \)
good2 \( 1 + (0.734 - 0.533i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 - 1.78iT - T^{2} \)
29 \( 1 + (0.0966 - 0.297i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 + 1.61iT - T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.183 - 0.253i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.309 + 0.951i)T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 + (-1.16 + 1.59i)T + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.690 + 0.951i)T + (-0.309 + 0.951i)T^{2} \)
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

−10.69672024028364157074166705647, −9.691169460784368002436702431350, −8.949765729037654034744893271477, −8.284580128500657370635364216275, −7.50340724305552595658435391941, −6.64546497063641141844514118203, −5.61202361112755444232928647323, −4.38839005943541075756168089498, −3.33878064382717655120915340626, −1.56068324131744474829727750197, 1.23470822574348972553277143840, 2.35405747009839158525114193277, 4.00680523639470954762419679314, 4.98461130327721379562703744967, 6.04606971751183770991396506099, 7.18288744374796686724519578318, 8.196570808172035148461442265304, 8.905522486581032259786156313159, 9.653174689899536106897026409875, 10.57349426378619545675900608310

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