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Properties

Label 1-115-115.42-r0-0-0
Degree $1$
Conductor $115$
Sign $0.958 - 0.285i$
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  + (−0.755 − 0.654i)2-s + (−0.540 − 0.841i)3-s + (0.142 + 0.989i)4-s + (−0.142 + 0.989i)6-s + (0.281 + 0.959i)7-s + (0.540 − 0.841i)8-s + (−0.415 + 0.909i)9-s + (0.654 + 0.755i)11-s + (0.755 − 0.654i)12-s + (−0.281 + 0.959i)13-s + (0.415 − 0.909i)14-s + (−0.959 + 0.281i)16-s + (0.989 + 0.142i)17-s + (0.909 − 0.415i)18-s + (−0.142 − 0.989i)19-s + ⋯
L(s)  = 1  + (−0.755 − 0.654i)2-s + (−0.540 − 0.841i)3-s + (0.142 + 0.989i)4-s + (−0.142 + 0.989i)6-s + (0.281 + 0.959i)7-s + (0.540 − 0.841i)8-s + (−0.415 + 0.909i)9-s + (0.654 + 0.755i)11-s + (0.755 − 0.654i)12-s + (−0.281 + 0.959i)13-s + (0.415 − 0.909i)14-s + (−0.959 + 0.281i)16-s + (0.989 + 0.142i)17-s + (0.909 − 0.415i)18-s + (−0.142 − 0.989i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.958 - 0.285i$
Analytic conductor: \(0.534057\)
Root analytic conductor: \(0.534057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (0:\ ),\ 0.958 - 0.285i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6119948935 - 0.08934791519i\)
\(L(\frac12)\) \(\approx\) \(0.6119948935 - 0.08934791519i\)
\(L(1)\) \(\approx\) \(0.6459102519 - 0.1593128158i\)
\(L(1)\) \(\approx\) \(0.6459102519 - 0.1593128158i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.755 - 0.654i)T \)
3 \( 1 + (-0.540 - 0.841i)T \)
7 \( 1 + (0.281 + 0.959i)T \)
11 \( 1 + (0.654 + 0.755i)T \)
13 \( 1 + (-0.281 + 0.959i)T \)
17 \( 1 + (0.989 + 0.142i)T \)
19 \( 1 + (-0.142 - 0.989i)T \)
29 \( 1 + (0.142 - 0.989i)T \)
31 \( 1 + (0.841 + 0.540i)T \)
37 \( 1 + (-0.909 - 0.415i)T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (0.540 + 0.841i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.281 - 0.959i)T \)
59 \( 1 + (0.959 + 0.281i)T \)
61 \( 1 + (-0.841 - 0.540i)T \)
67 \( 1 + (0.755 + 0.654i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (-0.989 + 0.142i)T \)
79 \( 1 + (-0.959 - 0.281i)T \)
83 \( 1 + (0.909 + 0.415i)T \)
89 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (0.909 - 0.415i)T \)
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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

−29.27801858887383441015358162625, −27.75864948017258687405458463050, −27.37893417692094706373692565753, −26.55049349506074770191812367356, −25.45651105143709454750614134212, −24.27603462537711231082527930844, −23.253452874610843686918229304206, −22.45448924885412503811992142400, −20.937699999388488812148621717757, −20.07689738390966992110820832623, −18.81636054123611730775780901317, −17.47413203541278210898868474547, −16.881232266217850524610191035105, −16.03091027454549305532744687471, −14.80883089731929808772817768907, −13.97398005230254751762086095074, −11.95353578369787069510940582026, −10.64006891934595623396927524954, −10.11540460056891225017362891089, −8.76097577199500675206035109816, −7.541401520487602384216026191609, −6.15592653456371090906610034642, −5.12110564594850043007729812475, −3.62287264422437595733836892273, −0.939642640389531348865610375234, 1.47960644087541399431025129898, 2.58996344912464413362063000997, 4.644962413479271333719540283896, 6.36505543039292810204524270765, 7.48021678280569522123133842202, 8.6853084353821995236507434740, 9.81801560349478697160255648063, 11.37536580233939529064963754690, 11.97012099616947001061446588682, 12.852152666882764661325884229197, 14.34530092497435447500944131440, 15.98001930305520920528246792457, 17.22695712562780192266769106943, 17.82560974500817123963468713654, 19.00969301052997505618239166460, 19.4805733301575431077011200579, 21.02727140921889829028172337998, 21.94167466617338508393132651438, 22.98185734078609132365062149831, 24.4047072672142860632697852731, 25.186581731470450266196903497562, 26.19656217365401881228960160214, 27.71018632553408850397562663665, 28.22846792457049228396049195449, 29.05534522664938802058481595414

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