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Properties

Label 1-184-184.117-r0-0-0
Degree $1$
Conductor $184$
Sign $0.182 - 0.983i$
Analytic cond. $0.854492$
Root an. cond. $0.854492$
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  + (0.959 − 0.281i)3-s + (−0.841 − 0.540i)5-s + (−0.142 − 0.989i)7-s + (0.841 − 0.540i)9-s + (−0.415 + 0.909i)11-s + (0.142 − 0.989i)13-s + (−0.959 − 0.281i)15-s + (−0.654 − 0.755i)17-s + (0.654 − 0.755i)19-s + (−0.415 − 0.909i)21-s + (0.415 + 0.909i)25-s + (0.654 − 0.755i)27-s + (0.654 + 0.755i)29-s + (−0.959 − 0.281i)31-s + (−0.142 + 0.989i)33-s + ⋯
L(s)  = 1  + (0.959 − 0.281i)3-s + (−0.841 − 0.540i)5-s + (−0.142 − 0.989i)7-s + (0.841 − 0.540i)9-s + (−0.415 + 0.909i)11-s + (0.142 − 0.989i)13-s + (−0.959 − 0.281i)15-s + (−0.654 − 0.755i)17-s + (0.654 − 0.755i)19-s + (−0.415 − 0.909i)21-s + (0.415 + 0.909i)25-s + (0.654 − 0.755i)27-s + (0.654 + 0.755i)29-s + (−0.959 − 0.281i)31-s + (−0.142 + 0.989i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(184\)    =    \(2^{3} \cdot 23\)
Sign: $0.182 - 0.983i$
Analytic conductor: \(0.854492\)
Root analytic conductor: \(0.854492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{184} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 184,\ (0:\ ),\ 0.182 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9857761970 - 0.8198411348i\)
\(L(\frac12)\) \(\approx\) \(0.9857761970 - 0.8198411348i\)
\(L(1)\) \(\approx\) \(1.120076143 - 0.4269480955i\)
\(L(1)\) \(\approx\) \(1.120076143 - 0.4269480955i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.959 - 0.281i)T \)
5 \( 1 + (-0.841 - 0.540i)T \)
7 \( 1 + (-0.142 - 0.989i)T \)
11 \( 1 + (-0.415 + 0.909i)T \)
13 \( 1 + (0.142 - 0.989i)T \)
17 \( 1 + (-0.654 - 0.755i)T \)
19 \( 1 + (0.654 - 0.755i)T \)
29 \( 1 + (0.654 + 0.755i)T \)
31 \( 1 + (-0.959 - 0.281i)T \)
37 \( 1 + (-0.841 + 0.540i)T \)
41 \( 1 + (0.841 + 0.540i)T \)
43 \( 1 + (0.959 - 0.281i)T \)
47 \( 1 + T \)
53 \( 1 + (0.142 + 0.989i)T \)
59 \( 1 + (0.142 - 0.989i)T \)
61 \( 1 + (0.959 + 0.281i)T \)
67 \( 1 + (-0.415 - 0.909i)T \)
71 \( 1 + (0.415 + 0.909i)T \)
73 \( 1 + (-0.654 + 0.755i)T \)
79 \( 1 + (-0.142 + 0.989i)T \)
83 \( 1 + (-0.841 + 0.540i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (0.841 + 0.540i)T \)
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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

−27.16878154105878912269784300697, −26.50727986515745434069561020056, −25.74708143273936725876992933358, −24.57070681579817655495512613115, −23.86245692731059948635466345416, −22.46631983200986813068175902617, −21.63139160215227910625548838813, −20.81414096890702843501262268980, −19.38385141557339912622894527736, −19.099322217175632134239069034715, −18.12279686005744765042584491265, −16.212552184638924054749635965488, −15.7380827323264007277950343086, −14.706869017062275554085316055593, −13.902236602493896168433493375400, −12.60489690849190803703098158367, −11.47954121852548855679638249254, −10.42544305443547579603894971205, −9.04340329639067978209367825218, −8.36524652102842523889085270834, −7.24933171079255302783017833528, −5.86672098680011359672216457218, −4.22176608079777327500653475432, −3.2584809962204450292862453417, −2.13630926417064765204534608629, 0.97404432135755038917229522891, 2.75348119186849971940807938153, 3.92324268924329853914821976700, 4.9472847044108418527662434165, 7.09395523904752262891447000902, 7.55953563200312368505100166364, 8.71060791992038585847355415803, 9.78677000558843040810102971046, 10.99618843404488732455619442619, 12.45700596162992146388471480526, 13.134617538024979547629649279196, 14.14994865552391579725538181901, 15.41060477516800201742539732771, 15.94223989936279650099795522231, 17.42646581739609810595496352219, 18.37390557567375703328370056876, 19.72130025673248189929348360437, 20.177417338600056435917669046646, 20.73245830565392341175971109179, 22.418009034914709782273717928811, 23.40713901506949878083156698210, 24.15273492341837058129279592699, 25.145278257836116480660866318561, 26.09984366414291255766958802953, 26.94686432746607898371537806257

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