L-function 1-115-115.29-r0-0-0

• Label: 1-115-115.29-r0-0-0
• Degree: $1$
• Conductor: $115$
• Sign: $0.117 + 0.993i$
• Analytic cond.: $0.534057$
• Root an. cond.: $0.534057$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.654 + 0.755i)2^-s + (−0.841 − 0.540i)3^-s + (−0.142 + 0.989i)4^-s + (−0.142 − 0.989i)6^-s + (0.959 + 0.281i)7^-s + (−0.841 + 0.540i)8^-s + (0.415 + 0.909i)9^-s + (−0.654 + 0.755i)11^-s + (0.654 − 0.755i)12^-s + (0.959 − 0.281i)13^-s + (0.415 + 0.909i)14^-s + (−0.959 − 0.281i)16^-s + (0.142 + 0.989i)17^-s + (−0.415 + 0.909i)18^-s + (−0.142 + 0.989i)19^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$115$    =    $5 \cdot 23$
Sign:	$0.117 + 0.993i$
Analytic conductor:	$0.534057$
Root analytic conductor:	$0.534057$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{115} (29, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 115,\ (0:\ ),\ 0.117 + 0.993i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.8585184344 + 0.7629518205i$
$L(1)$	$\approx$	$1.035359787 + 0.5147159243i$

Euler product

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

	$p$	$F_p(T)$
bad	5	$1$
	23	$1$
good	2	$1 + (0.654 + 0.755i)T$
	3	$1 + (-0.841 - 0.540i)T$
	7	$1 + (0.959 + 0.281i)T$
	11	$1 + (-0.654 + 0.755i)T$
	13	$1 + (0.959 - 0.281i)T$
	17	$1 + (0.142 + 0.989i)T$
	19	$1 + (-0.142 + 0.989i)T$
	29	$1 + (-0.142 - 0.989i)T$
	31	$1 + (0.841 - 0.540i)T$

Imaginary part of the first few zeros on the critical line

−29.10313705408449113602438256313, −28.12958585129910008323113886517, −27.40179290352737298892919125166, −26.372400029302204110033436822059, −24.46135446751767404938751774427, −23.642101437971210703475406259129, −22.93549466958205247821651350339, −21.63377500692663553591240079733, −21.138165870545601882848428075997, −20.17532366408516165252220401144, −18.57112217570843081551417639400, −17.86316426863556526511640041436, −16.34016731878351938196561268028, −15.381599644849654167574752922932, −14.12136715818564179379916440548, −13.10753359924897240147158079461, −11.57767757730299468987149450972, −11.161629491719913628999987854210, −10.11672381704947753976617204399, −8.68780761877440654332002743930, −6.672018228862258725994163175385, −5.33716758378048650157559078796, −4.57234444883937206243247490292, −3.14839730646965663572739337693, −1.13747561405003264459911904188, 1.9884665858319784473709972679, 4.1239627441830562950890759559, 5.35106421411417604257955237406, 6.17661018720641704698213896638, 7.59679018249535108016140936049, 8.3392594685906112067006529222, 10.48719988525456163466462478897, 11.71431778442403459623363903244, 12.62296673249292294839742534713, 13.60340054532178519259264743249, 14.900491362441900189552512892834, 15.85741707899964335477538851776, 17.10897088402198095512940628583, 17.82266348041846248770362500590, 18.74253248074542819637869969331, 20.72089029193433773995731298477, 21.46583805903655275502911473533, 22.80697555371590991582955427974, 23.34961716492423017552901325337, 24.32719194328124229375103689521, 25.12612126893441070904139643517, 26.24344291348786609789518827953, 27.633278141626577538394213944957, 28.40067718982267396478890905375, 29.84356158636209520110215730914

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