L-function 1-693-693.139-r0-0-0

• Label: 1-693-693.139-r0-0-0
• Degree: $1$
• Conductor: $693$
• Sign: $0.994 + 0.104i$
• 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

+ (0.978 + 0.207i)2^-s + (0.913 + 0.406i)4^-s + (0.978 − 0.207i)5^-s + (0.809 + 0.587i)8^-s + 10^-s + (0.669 − 0.743i)13^-s + (0.669 + 0.743i)16^-s + (0.309 − 0.951i)17^-s + (−0.809 − 0.587i)19^-s + (0.978 + 0.207i)20^-s + (−0.5 − 0.866i)23^-s + (0.913 − 0.406i)25^-s + (0.809 − 0.587i)26^-s + (0.104 + 0.994i)29^-s + (−0.669 + 0.743i)31^-s + (0.5 + 0.866i)32^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$693$    =    $3^{2} \cdot 7 \cdot 11$
Sign:	$0.994 + 0.104i$
Analytic conductor:	$3.21827$
Root analytic conductor:	$3.21827$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{693} (139, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 693,\ (0:\ ),\ 0.994 + 0.104i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$3.287685947 + 0.1727612488i$
$L(1)$	$\approx$	$2.239908734 + 0.1498800839i$

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	$1 + (0.978 + 0.207i)T$
	5	$1 + (0.978 - 0.207i)T$
	13	$1 + (0.669 - 0.743i)T$
	17	$1 + (0.309 - 0.951i)T$
	19	$1 + (-0.809 - 0.587i)T$
	23	$1 + (-0.5 - 0.866i)T$
	29	$1 + (0.104 + 0.994i)T$
	31	$1 + (-0.669 + 0.743i)T$
	37	$1 + (-0.809 + 0.587i)T$

Imaginary part of the first few zeros on the critical line

−22.61920750891218258564063429400, −21.70047030687649263879490739140, −21.2136746050238482616072355556, −20.620654590183846511167446886712, −19.39100159779084354901046722623, −18.845067293683744795553754145896, −17.66803540190110700296226725425, −16.829817479365749513507968553143, −16.00056572726707846687837726957, −14.949795461943155130209900790235, −14.31668483782519872116003670044, −13.51235673966195358097256339602, −12.89724155704829269575139036602, −11.91454329581411890198408999817, −10.9789727454255128584350248040, −10.24667250094860138544002908246, −9.37576869988910668437888195746, −8.12150512639492419447338250749, −6.90919325758338002686504466672, −6.06222396088647521518425905760, −5.56490947626119868376857475701, −4.23844682624331233575880471031, −3.5015514728284213711627251161, −2.14315995232039463604823977777, −1.59362849958098197500200156460, 1.35365301087204146301922415931, 2.49586391665703685077596615639, 3.34018292329332334622079727905, 4.653409363711673107499457547779, 5.33757814954631043660791864533, 6.246022399861904464898497471716, 6.96705475978490963582442436645, 8.18169062350160141322527747987, 9.08089093779208639666617378059, 10.35815331361592414388180495934, 10.922339716859829627157317483241, 12.17332005890289892471083778253, 12.836524328576352071004935914167, 13.623241167487482717416854902097, 14.26026242490766357674297985034, 15.145651468506053160247113495648, 16.10498974686826102229692907854, 16.73807330422269540532663887086, 17.71552760090339120472703019895, 18.42743746255005525965216705649, 19.80021655425968087842039881997, 20.525267695231047100249146340991, 21.12837012188667356935544281267, 21.96939696227396878891518879857, 22.58254821811555984885104301275

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