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Properties

Label 1-684-684.187-r1-0-0
Degree $1$
Conductor $684$
Sign $0.934 + 0.356i$
Analytic cond. $73.5060$
Root an. cond. $73.5060$
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  + (0.766 − 0.642i)5-s − 7-s + (0.5 − 0.866i)11-s + (−0.939 + 0.342i)13-s + (0.173 + 0.984i)17-s + (−0.173 + 0.984i)23-s + (0.173 − 0.984i)25-s + (−0.939 + 0.342i)29-s + (0.5 + 0.866i)31-s + (−0.766 + 0.642i)35-s + 37-s + (0.173 + 0.984i)41-s + (−0.173 − 0.984i)43-s + (0.939 − 0.342i)47-s + 49-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)5-s − 7-s + (0.5 − 0.866i)11-s + (−0.939 + 0.342i)13-s + (0.173 + 0.984i)17-s + (−0.173 + 0.984i)23-s + (0.173 − 0.984i)25-s + (−0.939 + 0.342i)29-s + (0.5 + 0.866i)31-s + (−0.766 + 0.642i)35-s + 37-s + (0.173 + 0.984i)41-s + (−0.173 − 0.984i)43-s + (0.939 − 0.342i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.934 + 0.356i$
Analytic conductor: \(73.5060\)
Root analytic conductor: \(73.5060\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 684,\ (1:\ ),\ 0.934 + 0.356i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.804819160 + 0.3323567311i\)
\(L(\frac12)\) \(\approx\) \(1.804819160 + 0.3323567311i\)
\(L(1)\) \(\approx\) \(1.089545122 - 0.04056889540i\)
\(L(1)\) \(\approx\) \(1.089545122 - 0.04056889540i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (0.766 - 0.642i)T \)
7 \( 1 - T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.939 + 0.342i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (-0.939 + 0.342i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (-0.939 + 0.342i)T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (-0.766 - 0.642i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (0.939 + 0.342i)T \)
83 \( 1 - T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (0.766 - 0.642i)T \)
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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

−22.590026830494110717321601866307, −21.915294285075885789952882216117, −20.73699526725404242323404114119, −20.09089053914751664812369091258, −19.0850474975742800123487775013, −18.45441580032623227736839534776, −17.47431612849452355250157413466, −16.87643187673891555955493840343, −15.8639030886621547933343098178, −14.873120266079394527934200697181, −14.29989859972261598155772931561, −13.25396251744700300138715402095, −12.57243432928999954896205303878, −11.62310330525067503873969699807, −10.481949266071290421828303894640, −9.627234752395311455979687404920, −9.39110919802331502872124720542, −7.73526156403429247640435352124, −6.91924865305509371248193332489, −6.219934744386076684747025415902, −5.19067819932268276755627749209, −4.031281784969665757758912443286, −2.799690063687865504059249163233, −2.191383472316973436611100194088, −0.540938792749134326283525265721, 0.83685204453732153558175188908, 1.97304052492939743014774061759, 3.16498569979064360203349542081, 4.157128146757576396005329517317, 5.41016791387498847766514463041, 6.08295499685769060661757917437, 6.977994608978443520093908115358, 8.223041521214476875332237064745, 9.191422091590191957627086223, 9.70742114828628587135381392299, 10.6511582626361808877355419133, 11.85832019321202299907397503997, 12.63531221958390082374484368845, 13.38345990843712820167699766882, 14.12745685564635462891602315453, 15.12137468439221783291142515120, 16.23037117495907211228063564806, 16.802853459443285592341486356941, 17.40733533584113234349349617994, 18.55710565872912093990517528822, 19.48197086191083957546889463465, 19.8924381848555555469772723999, 21.10185155683289377632600152292, 21.852077881087536119224140957696, 22.220004213588845481796870625581

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