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Properties

Label 1-92-92.15-r0-0-0
Degree $1$
Conductor $92$
Sign $0.529 - 0.848i$
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  + (0.654 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.529 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.529 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $0.529 - 0.848i$
Analytic conductor: \(0.427246\)
Root analytic conductor: \(0.427246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{92} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 92,\ (0:\ ),\ 0.529 - 0.848i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.061752144 - 0.5892415556i\)
\(L(\frac12)\) \(\approx\) \(1.061752144 - 0.5892415556i\)
\(L(1)\) \(\approx\) \(1.168571365 - 0.3988168022i\)
\(L(1)\) \(\approx\) \(1.168571365 - 0.3988168022i\)

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

−30.795106063783915902594235561211, −29.584521099411673939731538312342, −28.408647059486744645756805332373, −26.80805101014496597192742430060, −26.62332079461077617596027325204, −25.666363257889887911242065448741, −24.21288268971400388150175225823, −23.038590492736754945785161224431, −21.905363016018520410127183197232, −21.002249226927515790900569281395, −20.01940238039879443168631221083, −18.823508108386155323461409754879, −17.70643734673537649137229845962, −16.24193747203072613137506235514, −15.34095639827908844727841077923, −14.00018265883928437487019280554, −13.63218580681171477165114361676, −11.27311281570759548725257694492, −10.59338793601727773134610896048, −9.42107165116403880562799973391, −7.97963675357926205589800191303, −6.8419086715596014732595110305, −4.96103933626250544064299173520, −3.65630477719136189554145260427, −2.3708199900890198514253246559, 1.50984416072749612545460215344, 2.90408157574745849165442649621, 4.89342566782833053586186581796, 6.14547950618245809659829913400, 8.01573352111948690496932249241, 8.486367546889064038382085299126, 9.88293503915710432484369458360, 11.7579440558038620016128101593, 12.76653992671027723839694143988, 13.51313457659889814148933536091, 15.014813865132945693216720928572, 15.91084019527906837991193756389, 17.67051522115026834071537387289, 18.24484145179155861631994273982, 19.62772595501735741122619959323, 20.569023916213222789878824596237, 21.37242449735258548692452443965, 23.05085135889571442489497774631, 24.19363233549228004448566827660, 24.86912340038040220788841208447, 25.68706110858232847552923221009, 27.03474674818034429491694343586, 28.411579962794133900056682953346, 28.94695516352197558087512426759, 30.436184063525679307130013491475

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