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Properties

Label 1-92-92.59-r1-0-0
Degree $1$
Conductor $92$
Sign $-0.293 + 0.956i$
Analytic cond. $9.88677$
Root an. cond. $9.88677$
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  + (−0.415 − 0.909i)3-s + (−0.654 − 0.755i)5-s + (−0.841 − 0.540i)7-s + (−0.654 + 0.755i)9-s + (0.142 + 0.989i)11-s + (0.841 − 0.540i)13-s + (−0.415 + 0.909i)15-s + (−0.959 + 0.281i)17-s + (0.959 + 0.281i)19-s + (−0.142 + 0.989i)21-s + (−0.142 + 0.989i)25-s + (0.959 + 0.281i)27-s + (−0.959 + 0.281i)29-s + (−0.415 + 0.909i)31-s + (0.841 − 0.540i)33-s + ⋯
L(s)  = 1  + (−0.415 − 0.909i)3-s + (−0.654 − 0.755i)5-s + (−0.841 − 0.540i)7-s + (−0.654 + 0.755i)9-s + (0.142 + 0.989i)11-s + (0.841 − 0.540i)13-s + (−0.415 + 0.909i)15-s + (−0.959 + 0.281i)17-s + (0.959 + 0.281i)19-s + (−0.142 + 0.989i)21-s + (−0.142 + 0.989i)25-s + (0.959 + 0.281i)27-s + (−0.959 + 0.281i)29-s + (−0.415 + 0.909i)31-s + (0.841 − 0.540i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.293 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.293 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $-0.293 + 0.956i$
Analytic conductor: \(9.88677\)
Root analytic conductor: \(9.88677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{92} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 92,\ (1:\ ),\ -0.293 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03205446198 + 0.04334947363i\)
\(L(\frac12)\) \(\approx\) \(0.03205446198 + 0.04334947363i\)
\(L(1)\) \(\approx\) \(0.5672425983 - 0.2168791443i\)
\(L(1)\) \(\approx\) \(0.5672425983 - 0.2168791443i\)

Euler product

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

−29.608544481102853695188230230434, −28.63279954212661400458995045577, −27.700566412867025369837643964358, −26.49116799978401800905189225479, −26.13502610550270559477679405449, −24.43687397181695660952779532195, −23.134713518296406579482894272737, −22.33206895520508819745119385713, −21.617181898221299098329583017704, −20.206021267896913765246048634074, −19.03873654024660254929937801529, −18.0753852777969686956855441839, −16.386473589331503213636175939929, −15.85992877714988686009452716042, −14.8093490553990374700818800893, −13.414593160753678873666088244941, −11.62612769056759329321838670279, −11.1338562455458090370177393947, −9.690469839225809713611447933875, −8.627390547527390057043442842458, −6.76035741145408786622345816478, −5.72846130675755650232921954377, −3.97608070231404472150270462634, −3.0234035064113433884954816974, −0.02666756667046684328547814893, 1.47065358511391874125696918336, 3.586343047280773912186090718401, 5.16778944999664416858687989329, 6.66150727616846477276422084060, 7.6460874684276345218335422520, 8.955719779541627995346621707078, 10.575644936459644887873465641881, 11.91118730720346920180904774764, 12.812825096627019922328254096984, 13.609214121069361117917912825517, 15.43762529017711407902316535888, 16.47417208806888063402178777919, 17.489360596258954437873666941311, 18.631339304605737655002903322989, 19.90269483475767145419761186946, 20.336342152685759454613026465808, 22.41130366954420493217085035257, 23.093763789451482768103706645371, 24.000985795523149731311371438480, 25.03762262745029541480491676139, 26.04575126863347457964098115264, 27.540502325081611666163510594766, 28.4932677616779296306365880177, 29.204784730390528400765976019166, 30.53273715745721181760654698313

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