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Properties

Label 1-693-693.32-r0-0-0
Degree $1$
Conductor $693$
Sign $0.805 + 0.592i$
Analytic cond. $3.21827$
Root an. cond. $3.21827$
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 − 0.866i)2-s + (−0.5 + 0.866i)4-s − 5-s + 8-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 − 0.866i)20-s − 23-s + 25-s + (0.5 − 0.866i)26-s + (−0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s + (−0.5 + 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s − 5-s + 8-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 − 0.866i)20-s − 23-s + 25-s + (0.5 − 0.866i)26-s + (−0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s + (−0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.805 + 0.592i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ 0.805 + 0.592i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5360900049 + 0.1760960544i\)
\(L(\frac12)\) \(\approx\) \(0.5360900049 + 0.1760960544i\)
\(L(1)\) \(\approx\) \(0.6080753037 - 0.1303986076i\)
\(L(1)\) \(\approx\) \(0.6080753037 - 0.1303986076i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 - T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.80305115797802716828628419357, −22.16213509781419241935135155536, −20.713830149929332418663206007171, −19.93185695073101569711190830966, −19.25891758769124511858845811437, −18.383771922264737380759448364706, −17.74742396666492745717528271428, −16.67060476106497601336525872322, −16.06152655737370681873114388374, −15.23682447206393764287489413754, −14.72930915386625787631631996017, −13.591169707331299699928850924913, −12.72673839827670040072952235899, −11.60383369740638368263471700558, −10.6996224236016427626769958689, −9.90200132088691840539954318677, −8.75394888390677208866140702503, −8.00714983490002207536317510877, −7.49391815320297455074610484768, −6.2751970621488979762235728752, −5.56355934888621978123952293046, −4.30825474984036596068154420183, −3.56312509962547293896148575124, −1.82642537722738189147419384378, −0.39410316643923769480489398755, 1.06317259218063687051983044075, 2.344706130246995665700016991786, 3.422701962272879002058319609319, 4.19398563763127786504041782536, 5.1357644321695903418898662779, 6.88442163316802297524589104362, 7.458321122587492295260099057745, 8.67579609503324411488824731409, 9.06502037415165436292620073327, 10.28959123679291261647935114664, 11.1836394193173492445059545660, 11.72969787418604211677801773820, 12.47713181538750838376137944, 13.548423521338536255765780726779, 14.26870769618285258163138234225, 15.71668098368948872672971989397, 16.12697814878184627478448189324, 17.15341446334154208355987932735, 18.16798113309083133434041662881, 18.71112313661937231657632917627, 19.58324838903521845378608061586, 20.234026157174003570859902467923, 20.8514125956368601348766731968, 22.07938357159106042202134399328, 22.39332208303111144897615043886

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