L-function 1-693-693.32-r0-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	3	$1$
	7	$1$
	11	$1$
good	2	$1 + (-0.5 - 0.866i)T$
	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	$1 + (-0.5 + 0.866i)T$

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