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Properties

Label 1-93-93.23-r0-0-0
Degree $1$
Conductor $93$
Sign $0.0525 + 0.998i$
Analytic cond. $0.431890$
Root an. cond. $0.431890$
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  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s − 5-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)10-s + (0.309 + 0.951i)11-s + (0.809 − 0.587i)13-s + (−0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)17-s + (−0.809 − 0.587i)19-s + (−0.309 − 0.951i)20-s + (−0.309 + 0.951i)22-s + (0.309 − 0.951i)23-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s − 5-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)10-s + (0.309 + 0.951i)11-s + (0.809 − 0.587i)13-s + (−0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)17-s + (−0.809 − 0.587i)19-s + (−0.309 − 0.951i)20-s + (−0.309 + 0.951i)22-s + (0.309 − 0.951i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0525 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0525 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.0525 + 0.998i$
Analytic conductor: \(0.431890\)
Root analytic conductor: \(0.431890\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 93,\ (0:\ ),\ 0.0525 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9782381678 + 0.9281326887i\)
\(L(\frac12)\) \(\approx\) \(0.9782381678 + 0.9281326887i\)
\(L(1)\) \(\approx\) \(1.202644646 + 0.6658986055i\)
\(L(1)\) \(\approx\) \(1.202644646 + 0.6658986055i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 - T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 - T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + (0.809 + 0.587i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−30.14890436982753007601308595579, −29.337074005970340900005553393562, −27.927654995891986742446422875769, −27.2602305833281809562448977389, −25.932229656431034208360948114801, −24.201079187263207767076125848679, −23.66967212516826229747218658683, −22.828282907040982759489286425855, −21.48201957037279086979732780652, −20.60291274004950308356703845729, −19.45228867468449102582393636072, −18.846635092319132867681865055498, −16.91636931563261481081097101910, −15.81746317334278001306744440233, −14.598905809076479871800947254062, −13.67183526420510332857306485372, −12.43999371410464506265174488843, −11.212849696832466897972327454307, −10.64163506488860945026731176311, −8.76039133740248175945812175545, −7.26894266026897109327844619209, −5.86691937682772208089113420988, −4.14085078041530229204433032287, −3.56419428996830271227761802187, −1.337164301210660044853081870010, 2.61350784459411592742076044349, 4.094585611232969422776071046362, 5.221665571838384525805213317176, 6.68804968489805835544211952249, 7.8817436202865462067016407300, 8.9276175806391470516547624184, 11.10205564286380500246871041991, 12.08084430478710381751845001639, 12.95004560292508977338778669109, 14.58065474927166684717111377317, 15.307995515443696091418335569523, 16.13622220002828122486254090057, 17.54805493033819584622572372606, 18.72104723631273324657235463439, 20.22838607319401959310375852852, 21.09414945295063846291512142256, 22.54249574519075862231088200189, 23.02508093084276483667213169390, 24.270345749508353371359040835142, 25.1152322646238317480482902495, 26.12224366491822319645525641238, 27.4835183363561622952919706946, 28.25927197397302585222928400450, 30.00867265895377296861066386542, 30.79124894015829289723596599561

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