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Properties

Label 1-92-92.75-r1-0-0
Degree $1$
Conductor $92$
Sign $-0.392 - 0.919i$
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  + (−0.841 − 0.540i)3-s + (0.415 − 0.909i)5-s + (0.959 + 0.281i)7-s + (0.415 + 0.909i)9-s + (0.654 − 0.755i)11-s + (−0.959 + 0.281i)13-s + (−0.841 + 0.540i)15-s + (−0.142 − 0.989i)17-s + (0.142 − 0.989i)19-s + (−0.654 − 0.755i)21-s + (−0.654 − 0.755i)25-s + (0.142 − 0.989i)27-s + (−0.142 − 0.989i)29-s + (−0.841 + 0.540i)31-s + (−0.959 + 0.281i)33-s + ⋯
L(s)  = 1  + (−0.841 − 0.540i)3-s + (0.415 − 0.909i)5-s + (0.959 + 0.281i)7-s + (0.415 + 0.909i)9-s + (0.654 − 0.755i)11-s + (−0.959 + 0.281i)13-s + (−0.841 + 0.540i)15-s + (−0.142 − 0.989i)17-s + (0.142 − 0.989i)19-s + (−0.654 − 0.755i)21-s + (−0.654 − 0.755i)25-s + (0.142 − 0.989i)27-s + (−0.142 − 0.989i)29-s + (−0.841 + 0.540i)31-s + (−0.959 + 0.281i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $-0.392 - 0.919i$
Analytic conductor: \(9.88677\)
Root analytic conductor: \(9.88677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{92} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 92,\ (1:\ ),\ -0.392 - 0.919i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7111909106 - 1.076729550i\)
\(L(\frac12)\) \(\approx\) \(0.7111909106 - 1.076729550i\)
\(L(1)\) \(\approx\) \(0.8628975937 - 0.4210481654i\)
\(L(1)\) \(\approx\) \(0.8628975937 - 0.4210481654i\)

Euler product

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

−30.1525683146824678227463085157, −29.5256242921364533651667027065, −28.21938888434475341821211739431, −27.27248269180289795591194532442, −26.54819565859175912219021329954, −25.208727876736604418699565770561, −23.94842494662172116922687374776, −22.85219433629511835448746630118, −22.02622087888949005983536434520, −21.161186460963174116872997545360, −19.86703926308540094621746252671, −18.23978053377389612595622251616, −17.54301399030236973431003812620, −16.649986022342650632012977709538, −14.84037501363039116968190860418, −14.6214660379216703616916134108, −12.64306980516852390151225968573, −11.43622514152918811879244828132, −10.49730682232510423790731945202, −9.569721712563948454136586678601, −7.61646625995718237255854537494, −6.40186722886582272152100549114, −5.11601592361277779822024887575, −3.80991530204000556504147132120, −1.74326981650460929127055213301, 0.675585747098607964481219342732, 2.07388394478194674423780893993, 4.68120245092933230636497717886, 5.46248851137550244832340106086, 6.91484084776923333973023220680, 8.31103128943902432729595089774, 9.569958817076489571476168376237, 11.29912879065779390543300740327, 11.94523330617818750317027361140, 13.21848910966914142698458696989, 14.26585914033969274248888660519, 15.97344549447729242779332008316, 17.05902208392097799704809704469, 17.682005704791671388645428867022, 18.92245498673701864873288691373, 20.17991571993627594194857762817, 21.501599054232700947390957466169, 22.21280000119547995127120368485, 23.79339508778249633081551960055, 24.42363049952314867664858846788, 25.07662948554512841904379044696, 26.994458306924141334277006690530, 27.7908425781104068294759989373, 28.8162605501141968180309984924, 29.58060805391190405893378701819

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