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Properties

Label 1-184-184.61-r1-0-0
Degree $1$
Conductor $184$
Sign $-0.746 + 0.665i$
Analytic cond. $19.7735$
Root an. cond. $19.7735$
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  + (0.654 − 0.755i)3-s + (−0.142 + 0.989i)5-s + (−0.415 − 0.909i)7-s + (−0.142 − 0.989i)9-s + (−0.959 + 0.281i)11-s + (−0.415 + 0.909i)13-s + (0.654 + 0.755i)15-s + (−0.841 + 0.540i)17-s + (0.841 + 0.540i)19-s + (−0.959 − 0.281i)21-s + (−0.959 − 0.281i)25-s + (−0.841 − 0.540i)27-s + (−0.841 + 0.540i)29-s + (−0.654 − 0.755i)31-s + (−0.415 + 0.909i)33-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)3-s + (−0.142 + 0.989i)5-s + (−0.415 − 0.909i)7-s + (−0.142 − 0.989i)9-s + (−0.959 + 0.281i)11-s + (−0.415 + 0.909i)13-s + (0.654 + 0.755i)15-s + (−0.841 + 0.540i)17-s + (0.841 + 0.540i)19-s + (−0.959 − 0.281i)21-s + (−0.959 − 0.281i)25-s + (−0.841 − 0.540i)27-s + (−0.841 + 0.540i)29-s + (−0.654 − 0.755i)31-s + (−0.415 + 0.909i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(184\)    =    \(2^{3} \cdot 23\)
Sign: $-0.746 + 0.665i$
Analytic conductor: \(19.7735\)
Root analytic conductor: \(19.7735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{184} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 184,\ (1:\ ),\ -0.746 + 0.665i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1207758422 + 0.3171576430i\)
\(L(\frac12)\) \(\approx\) \(0.1207758422 + 0.3171576430i\)
\(L(1)\) \(\approx\) \(0.8791162419 - 0.05822846133i\)
\(L(1)\) \(\approx\) \(0.8791162419 - 0.05822846133i\)

Euler product

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

−26.70469041722334028441934624278, −25.61525617836405719732586599273, −24.84756634644563129579313310365, −24.03087195433408980980781486427, −22.507616172586741574373674318760, −21.81375987631263824359448468422, −20.69205920482364430516793031127, −20.140627286473516618202395692914, −19.09581144252154440736415368107, −17.90859680545434918139646236808, −16.57596353428814410304738365093, −15.64719763892656398371133580752, −15.28708635431823695577563022677, −13.65053458363878057536288213265, −12.92686366627847155261441683375, −11.71670950927529384359718466247, −10.36914166567781893308095046108, −9.29099623771565466791585559693, −8.61200432268903946410826629581, −7.54053815192814140258577653524, −5.504030703299592189995358149594, −4.926618502554073899905864200347, −3.36796164889890981272998814992, −2.30153775035417620730931597659, −0.10018556581416366125887146903, 1.84024519532557870129290746106, 3.02241636925054248028893831602, 4.12350256808333573735417374564, 6.10599893722907343318068468887, 7.21287980622247860112112856595, 7.6593621321682012600048846169, 9.24399798286677586454149256981, 10.31705292747361624998535385115, 11.4110697398747055490005888985, 12.73493782719377852359443834516, 13.62270486263031669956464828742, 14.446760087270630319205069384884, 15.39524640599707182063653266732, 16.71424156084143590447757998733, 18.00056146866754057850150244216, 18.647312990050125275705055687, 19.63929976265971767291153139307, 20.35123362196446606850576793612, 21.615722532920186680313307344462, 22.81279236315801803748670948536, 23.58230932568272067397421299905, 24.408223288984681286063059451312, 25.74340877679510054455786331980, 26.36246649117244922511362203096, 26.820665532519371141909828149814

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