L-function 1-680-680.523-r0-0-0

• Label: 1-680-680.523-r0-0-0
• Degree: $1$
• Conductor: $680$
• Sign: $0.346 - 0.937i$
• Analytic cond.: $3.15790$
• Root an. cond.: $3.15790$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

− 3^-s + 7^-s + 9^-s − i·11^-s − i·13^-s + 19^-s − 21^-s − 23^-s − 27^-s + i·29^-s − i·31^-s + i·33^-s − 37^-s + i·39^-s − i·41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.346 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$680$    =    $2^{3} \cdot 5 \cdot 17$
Sign:	$0.346 - 0.937i$
Analytic conductor:	$3.15790$
Root analytic conductor:	$3.15790$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{680} (523, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 680,\ (0:\ ),\ 0.346 - 0.937i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.8557475851 - 0.5958610731i$
$L(1)$	$\approx$	$0.8630672070 - 0.1765596926i$

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	$1 + T$
	7	$1 - T$
	11	$1$
	13	$1$
	19	$1 + T$
	23	$1$
	29	$1 + T$
	31	$1$
	37	$1 - iT$

Imaginary part of the first few zeros on the critical line

−22.96014151696852588850978412161, −22.0942237331739192614473374049, −21.32971135664219133508019034402, −20.64223904547135113784826962660, −19.62331470289611221717976337989, −18.49830584085125412789377762427, −17.88285309617003531883504647420, −17.31810287256225906163809701917, −16.34729600496306585412734490470, −15.62799278789867903229937583135, −14.61205712390440944450727013466, −13.81465909332051450762672829591, −12.687400080197835993458826892439, −11.71779537944226917809523704453, −11.49721093570378506892979154248, −10.23809555700394482810531565967, −9.62993760547918011452212400720, −8.31975945588761205346904361196, −7.334101647371517776735455985164, −6.611107362415511184890233470569, −5.440143390274081317987755458359, −4.727975978066337814811865364839, −3.92645685062218468108535244874, −2.12459115958700312178588790839, −1.2977819313864929308072407307, 0.646802113809041213200469767749, 1.75649262792744364828046318844, 3.26650895151819965216094454140, 4.385499032722467615170573850057, 5.47170226533864130231643879785, 5.81975703395037801057127380027, 7.19888589876619274988111153646, 7.934905300993800549318133337849, 8.96432234876426986313024739773, 10.253455052599813014955778782534, 10.82469029860791477435143166705, 11.6725872161654926481817609652, 12.31526581766121362825059189275, 13.44235980073667610826733161836, 14.20027867933164282301550533911, 15.34273070906546484682028846513, 16.03465500654641625044889465277, 16.90714853668755338672016189065, 17.73193611299030525737456357912, 18.23601947146395996940963344490, 19.11492962846611725711568610530, 20.350633655282052280782678369077, 20.95810503715195109780600773228, 22.075876511875223474831007435385, 22.31292396285236000886424856476

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