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Properties

Label 1-92-92.67-r0-0-0
Degree $1$
Conductor $92$
Sign $0.603 - 0.797i$
Analytic cond. $0.427246$
Root an. cond. $0.427246$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $0.603 - 0.797i$
Analytic conductor: \(0.427246\)
Root analytic conductor: \(0.427246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{92} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 92,\ (0:\ ),\ 0.603 - 0.797i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.052693461 - 0.5230916014i\)
\(L(\frac12)\) \(\approx\) \(1.052693461 - 0.5230916014i\)
\(L(1)\) \(\approx\) \(1.158938073 - 0.3282804012i\)
\(L(1)\) \(\approx\) \(1.158938073 - 0.3282804012i\)

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

−30.63800291930230603599913588227, −29.84554874308413423191424745570, −27.9039638181088630151767381435, −27.56293728443534629072756999133, −26.1989018741682699447457112525, −25.40041588492542700208753138747, −24.47453098252050672037785083867, −22.927374551973412009156383046151, −22.11006617757496805147194976890, −20.8194110998383949810341910741, −19.77334134873459374816124496606, −18.980123836652567791872630270889, −17.87944051556311139537586095991, −16.02854279309072161469140498366, −15.16605980952823200132606482355, −14.572604671856481472438893725085, −12.93506811271202364590172299218, −11.860967130189130499588812936908, −10.36101019248434890911498158096, −9.17714390522509692533751275023, −8.02814049509467313424326745768, −6.933732154875911559495820390485, −4.96583305017656594098673035637, −3.48669797600428629560699980921, −2.39901222815915689651926937583, 1.37028465976861361995939737996, 3.47968510353074205927520412811, 4.28380287513400049901367415706, 6.52454509048814206656599729525, 7.80734594671441136295380839109, 8.62825196545670279320886913461, 9.98871709470280843640310066738, 11.53956527015856874942615416170, 12.76919556884292656848107962103, 13.8549138051860017054194535924, 14.805195878262703275952710000420, 16.20349649702039831563362635446, 17.03788934275555513123247364555, 19.04688753519103438251309272590, 19.33023085270019436213122574605, 20.52819520147669204312579419077, 21.3926985221089192380174409648, 23.17381986185276647607332743388, 23.9648389267498432663528767577, 24.82371276450114758131951887993, 26.25553067938281263084995248382, 26.81966794746888076597140598676, 27.95079599857500972650947213955, 29.45970992529225072799506953066, 30.22855255089469165284182114468

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