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Properties

Label 1-115-115.88-r0-0-0
Degree $1$
Conductor $115$
Sign $-0.172 + 0.984i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.172 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.172 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.172 + 0.984i$
Analytic conductor: \(0.534057\)
Root analytic conductor: \(0.534057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (0:\ ),\ -0.172 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.096302453 + 1.305427075i\)
\(L(\frac12)\) \(\approx\) \(1.096302453 + 1.305427075i\)
\(L(1)\) \(\approx\) \(1.327485135 + 0.9524792804i\)
\(L(1)\) \(\approx\) \(1.327485135 + 0.9524792804i\)

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

−29.15987284980149558055706685257, −28.47013047779341164843070751081, −27.08405018897708147110134222332, −25.665956179323149425791638970986, −24.633364481495453220760586670280, −24.0462531623284441465207482565, −22.80534816775457553646773874162, −21.77762446843044094182461921136, −20.84074852410254338447192344641, −19.55835034635672472301615287077, −19.01231149888661926790079442513, −18.06456155285740021104226956717, −16.22533400073967441177366389058, −14.90069447753620050608266154436, −14.072244581184361779319569716672, −13.06269319154859398727435407656, −12.0881701639569503849925021381, −11.24978924939843135008809016934, −9.444841569757958871875124321546, −8.568042709177263631749114035649, −6.64684202484383240042714193801, −5.8972028420752934665901387604, −4.028745047258652043506470894406, −2.75564096274907657751657214845, −1.581981417835774547265523715893, 2.73650659871433570460212993716, 4.012998955306845276900652955, 4.80460818146367243207747519598, 6.46507095765335245585825446458, 7.64751020405258639263030247997, 8.87236459469221358408858680182, 10.18326863054071338417378072000, 11.45025930605688427089256162878, 13.095103378235224895879809483754, 13.79838932105669492951194856630, 15.02683539828321720364555277183, 15.6574649916933962158965245838, 16.85616228545197675318165667236, 17.66507691922650845555884611817, 19.85825349558143224269094579061, 20.30403530033552153291835830416, 21.54747969619697107131290246924, 22.52492517467040131220948081103, 23.2092148719247993682317319628, 24.653994572296354415372745917672, 25.48532075967508763095188535353, 26.41450577599437543637839246900, 27.13437750860762792452474674811, 28.44994122793916477370051747433, 30.110145591484343899334742113598

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