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Properties

Label 1-152-152.141-r1-0-0
Degree $1$
Conductor $152$
Sign $-0.0977 + 0.995i$
Analytic cond. $16.3346$
Root an. cond. $16.3346$
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  + (−0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + 7-s + (−0.5 + 0.866i)9-s − 11-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)21-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + 27-s + (−0.5 + 0.866i)29-s − 31-s + (0.5 + 0.866i)33-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + 7-s + (−0.5 + 0.866i)9-s − 11-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)21-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + 27-s + (−0.5 + 0.866i)29-s − 31-s + (0.5 + 0.866i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.0977 + 0.995i$
Analytic conductor: \(16.3346\)
Root analytic conductor: \(16.3346\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 152,\ (1:\ ),\ -0.0977 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6702831590 + 0.7393336329i\)
\(L(\frac12)\) \(\approx\) \(0.6702831590 + 0.7393336329i\)
\(L(1)\) \(\approx\) \(0.8784852387 + 0.08627270793i\)
\(L(1)\) \(\approx\) \(0.8784852387 + 0.08627270793i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−27.645893230036051767835912625124, −26.731161797196966770617689823027, −25.68622429885691521355593399258, −24.38333077692188520478889293701, −23.770944389785628160334843677970, −22.455025535917180697928606231743, −21.48495231367675016982623508617, −20.77172412105112433518634813188, −20.04231179222773193641543487647, −18.15892642534942719142997136472, −17.457074808352611804473731913463, −16.595449427373586071527635737618, −15.48545303141122063154256537242, −14.61389039005498075024599960797, −13.17432663969937261120643316902, −12.1802747400484667116197414974, −10.87873362700157441945969091570, −10.11931507413839285510109711777, −8.84924496923572368691635071743, −7.86903995391234338848912983671, −5.87459184385586371997939686829, −5.12958050884839852194235039923, −4.152745591535586326720891717137, −2.21891303244719861833974789665, −0.38265996194905344818226957430, 1.66345623240916865804959961507, 2.66164525598229018735477818742, 4.80773770484773524438272910154, 5.866082221427460626793246200764, 7.13373696189748915811067560762, 7.8285565242584904344911801114, 9.486298687036911477874182228208, 10.968010947447759195067924612, 11.43021572382668854596420530275, 12.85116071379159551357267051154, 13.90099363797751997015944483973, 14.63789659037557292210955200821, 16.14129526409035133335893654100, 17.42395661087770732086658788153, 18.13743811806040214757061947144, 18.72403791186959078441133711206, 20.05186417128156877020876717678, 21.38826951733716546126248662252, 22.135597982439202327457333437042, 23.3536352158540994988740311351, 24.02705379396665348118788829277, 25.019649961628795226401989110024, 26.05318154751820883650400859799, 27.0350567918198331997344419043, 28.21300298767405599784603495293

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