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Properties

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 $0$

Origins

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Normalization:  

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 + ⋯
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

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

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

\(L(\frac{1}{2})\) \(\approx\) \(0.8585184344 + 0.7629518205i\)
\(L(\frac12)\) \(\approx\) \(0.8585184344 + 0.7629518205i\)
\(L(1)\) \(\approx\) \(1.035359787 + 0.5147159243i\)
\(L(1)\) \(\approx\) \(1.035359787 + 0.5147159243i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.654 + 0.755i)T \)
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 \( 1 + (0.841 - 0.540i)T \)
37 \( 1 + (-0.415 - 0.909i)T \)
41 \( 1 + (0.415 - 0.909i)T \)
43 \( 1 + (-0.841 - 0.540i)T \)
47 \( 1 - T \)
53 \( 1 + (0.959 + 0.281i)T \)
59 \( 1 + (-0.959 + 0.281i)T \)
61 \( 1 + (0.841 - 0.540i)T \)
67 \( 1 + (0.654 + 0.755i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.142 - 0.989i)T \)
79 \( 1 + (-0.959 + 0.281i)T \)
83 \( 1 + (-0.415 - 0.909i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (-0.415 + 0.909i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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