Properties

Label 1-115-115.28-r0-0-0
Degree $1$
Conductor $115$
Sign $-0.107 - 0.994i$
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  + (0.540 − 0.841i)2-s + (0.989 − 0.142i)3-s + (−0.415 − 0.909i)4-s + (0.415 − 0.909i)6-s + (−0.755 − 0.654i)7-s + (−0.989 − 0.142i)8-s + (0.959 − 0.281i)9-s + (−0.841 + 0.540i)11-s + (−0.540 − 0.841i)12-s + (0.755 − 0.654i)13-s + (−0.959 + 0.281i)14-s + (−0.654 + 0.755i)16-s + (0.909 + 0.415i)17-s + (0.281 − 0.959i)18-s + (0.415 + 0.909i)19-s + ⋯
L(s)  = 1  + (0.540 − 0.841i)2-s + (0.989 − 0.142i)3-s + (−0.415 − 0.909i)4-s + (0.415 − 0.909i)6-s + (−0.755 − 0.654i)7-s + (−0.989 − 0.142i)8-s + (0.959 − 0.281i)9-s + (−0.841 + 0.540i)11-s + (−0.540 − 0.841i)12-s + (0.755 − 0.654i)13-s + (−0.959 + 0.281i)14-s + (−0.654 + 0.755i)16-s + (0.909 + 0.415i)17-s + (0.281 − 0.959i)18-s + (0.415 + 0.909i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.107 - 0.994i$
Analytic conductor: \(0.534057\)
Root analytic conductor: \(0.534057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (0:\ ),\ -0.107 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.080714052 - 1.203932226i\)
\(L(\frac12)\) \(\approx\) \(1.080714052 - 1.203932226i\)
\(L(1)\) \(\approx\) \(1.292090645 - 0.8710866815i\)
\(L(1)\) \(\approx\) \(1.292090645 - 0.8710866815i\)

Euler product

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

−29.9117743706099284155122112981, −28.49522322839824027655339226173, −27.11692643904601036730132131776, −26.035805675236686922145044634402, −25.69016104084876182664939937044, −24.54389846043861478072089867580, −23.66672242505453264934387454566, −22.38015153411735323780440880978, −21.417012094855110993599840183517, −20.621945840226299238185495616701, −19.00050708785757794351438590097, −18.357467414650447358208419264678, −16.550882515495520978859108166071, −15.78269582858264990085333555064, −14.98741025831790030238174228979, −13.64254505761390653925220222155, −13.18424444385115859907524696549, −11.72910314192758358895762694228, −9.75995155480842915429678266988, −8.77709714311247151007089944717, −7.779393368580819740660666617258, −6.48738529115959306888707366897, −5.17469910497437868199765869492, −3.64154659790839762031299623903, −2.683020104500789345828229770363, 1.49138281670151486374539702654, 3.08874423632654308614938348207, 3.819064797792719766422887719252, 5.49583561726702936724440563976, 7.1287968388849501330942616627, 8.51670982661109412957832288490, 9.94929626808068694852672356315, 10.47880812134934656395139925920, 12.3792759928082878873842606280, 13.07979161826336629899816377912, 13.996453520781944297740187344632, 15.05037806938861799737974062852, 16.16122967394435830842089677739, 18.1085427041094559942829141071, 18.91337794457542750445889721115, 20.04498964488253435354042320291, 20.56320054337773063915075217450, 21.58428857902324058045819491510, 23.0142785265851386434938630851, 23.56252348112538679251992599845, 25.027281782315803148926601604729, 25.960670741205725245719198681241, 27.01248050715662813079288902400, 28.16614495518225499822124839457, 29.31970240165594619884285690319

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