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Properties

Label 1-115-115.57-r0-0-0
Degree $1$
Conductor $115$
Sign $-0.0479 + 0.998i$
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  + (0.909 + 0.415i)2-s + (−0.281 + 0.959i)3-s + (0.654 + 0.755i)4-s + (−0.654 + 0.755i)6-s + (0.989 + 0.142i)7-s + (0.281 + 0.959i)8-s + (−0.841 − 0.540i)9-s + (−0.415 − 0.909i)11-s + (−0.909 + 0.415i)12-s + (−0.989 + 0.142i)13-s + (0.841 + 0.540i)14-s + (−0.142 + 0.989i)16-s + (0.755 + 0.654i)17-s + (−0.540 − 0.841i)18-s + (−0.654 − 0.755i)19-s + ⋯
L(s)  = 1  + (0.909 + 0.415i)2-s + (−0.281 + 0.959i)3-s + (0.654 + 0.755i)4-s + (−0.654 + 0.755i)6-s + (0.989 + 0.142i)7-s + (0.281 + 0.959i)8-s + (−0.841 − 0.540i)9-s + (−0.415 − 0.909i)11-s + (−0.909 + 0.415i)12-s + (−0.989 + 0.142i)13-s + (0.841 + 0.540i)14-s + (−0.142 + 0.989i)16-s + (0.755 + 0.654i)17-s + (−0.540 − 0.841i)18-s + (−0.654 − 0.755i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0479 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0479 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.0479 + 0.998i$
Analytic conductor: \(0.534057\)
Root analytic conductor: \(0.534057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (0:\ ),\ -0.0479 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.116692816 + 1.171528278i\)
\(L(\frac12)\) \(\approx\) \(1.116692816 + 1.171528278i\)
\(L(1)\) \(\approx\) \(1.320678253 + 0.8310275058i\)
\(L(1)\) \(\approx\) \(1.320678253 + 0.8310275058i\)

Euler product

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

−29.45270747738259189833815414882, −28.2676128936739806158766350906, −27.36175908101658298143126456290, −25.45660246605983476615416197462, −24.73950257325635268769158309197, −23.69033755253451113004848363423, −23.13095756977851525258267935313, −21.97009206698231468771253623133, −20.75993376375451594784780229194, −19.93185109884281910734157169976, −18.74207215456022967592649285111, −17.76614367210378632270591446269, −16.52787286032235917176708543095, −14.82223555483765360891388322579, −14.25816085253449625151683391733, −12.89850115997317156497221769070, −12.18242174650650638200478337137, −11.193128866614073508431578505171, −9.98494522268232651319959879302, −7.89962884379018939823946668318, −7.01207536741465946287539253803, −5.53408575985564710845972598428, −4.6153613327666109926390504793, −2.64921694258128042252712902807, −1.51096438911267002013531343642, 2.58800323923188025070761256102, 4.04621053333434988868265966005, 5.071739381372297013537383631887, 5.97242449274484072615688041326, 7.671211258268695334344914802436, 8.82718565059850219680840127111, 10.58402333583767976439879079313, 11.415090844678610917108585126213, 12.53686183337826379177598905318, 14.11078315044400639111473591434, 14.79770826936498517551219641169, 15.78540763149402097853891778815, 16.84650456648387401895822816992, 17.63093905900362833218794838136, 19.50514017079002749515793145096, 20.89411936917295020405878636510, 21.45395412519994789765747962178, 22.19841684877020345464429114789, 23.540123675391051504695091345706, 24.141788842876658754397933201999, 25.4396055379710132696272647581, 26.54161131516670992324722072671, 27.3145471270854826330786300301, 28.58321009438530936423483868423, 29.67359261648902098357932433213

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