L-function 1-1700-1700.1639-r0-0-0

• Label: 1-1700-1700.1639-r0-0-0
• Degree: $1$
• Conductor: $1700$
• Sign: $0.360 - 0.932i$
• Analytic cond.: $7.89476$
• Root an. cond.: $7.89476$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.233 + 0.972i)3^-s + (−0.923 − 0.382i)7^-s + (−0.891 − 0.453i)9^-s + (0.760 + 0.649i)11^-s + (−0.951 + 0.309i)13^-s + (−0.987 − 0.156i)19^-s + (0.587 − 0.809i)21^-s + (0.760 + 0.649i)23^-s + (0.649 − 0.760i)27^-s + (−0.972 − 0.233i)29^-s + (0.852 + 0.522i)31^-s + (−0.809 + 0.587i)33^-s + (−0.760 + 0.649i)37^-s + (−0.0784 − 0.996i)39^-s + (0.0784 − 0.996i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1700$    =    $2^{2} \cdot 5^{2} \cdot 17$
Sign:	$0.360 - 0.932i$
Analytic conductor:	$7.89476$
Root analytic conductor:	$7.89476$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1700} (1639, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1700,\ (0:\ ),\ 0.360 - 0.932i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.3413216164 - 0.2340258218i$
$L(1)$	$\approx$	$0.6914678248 + 0.1787922781i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
	17	$1$
good	3	$1 + (-0.233 + 0.972i)T$
	7	$1 + (-0.923 - 0.382i)T$
	11	$1 + (0.760 + 0.649i)T$
	13	$1 + (-0.951 + 0.309i)T$
	19	$1 + (-0.987 - 0.156i)T$
	23	$1 + (0.760 + 0.649i)T$
	29	$1 + (-0.972 - 0.233i)T$
	31	$1 + (0.852 + 0.522i)T$
	37	$1 + (-0.760 + 0.649i)T$

Imaginary part of the first few zeros on the critical line

−20.1257460868349441452535080968, −19.62507398793495566718669790384, −18.85588897138095028162319193792, −18.59926818406715312137185408930, −17.300442995645648552594824060647, −16.98183866801327271985969310225, −16.24685026773965913342184687497, −15.10079228566983729597009621560, −14.53119176436091592169628802629, −13.55586989957347399010745381109, −12.906649202384899799217569557850, −12.32142450465148916652720316325, −11.62411557563932908312044110238, −10.7627435010386577694430037849, −9.829830209899615426717277443984, −8.8974226028720781158113471780, −8.3061188362397746379461302656, −7.20552259814825661331344326582, −6.62445326375024025436817202031, −5.93470835678968370695977818907, −5.12621831259580404617815328886, −3.87474026598842209394509444570, −2.872637869479108132578247460167, −2.18689622854453391135267640658, −0.96187442681448025595197468084, 0.17652049321092819175152676399, 1.77855478746982932395809979441, 2.97566142863318156808798710779, 3.72017547669794129139339222518, 4.540730698874300231681605585031, 5.209540653189232914800176073165, 6.44078402422867447080687302400, 6.81322584544963210666762781707, 7.98080768083807779701347073036, 9.18384265807631805078264917686, 9.49727292556913951510523288750, 10.252575970724770050461985668428, 11.01814743421726564472853835129, 11.91322815463330605669280981947, 12.56240754236552335518847904456, 13.499609912825210194782854910129, 14.43801911217661283738121700189, 15.04975798319414783179882105226, 15.70649345596263180267280909289, 16.585022357076112984078941658384, 17.17893616999068171814878236214, 17.537089881171102849448672067141, 19.02282428271426367670039081523, 19.47800038350975230800088492319, 20.18695669678544069869211137833

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