Properties

Label 1-19e2-361.229-r0-0-0
Degree $1$
Conductor $361$
Sign $0.472 - 0.881i$
Analytic cond. $1.67647$
Root an. cond. $1.67647$
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  + (0.245 + 0.969i)2-s + (−0.0825 + 0.996i)3-s + (−0.879 + 0.475i)4-s + (0.546 − 0.837i)5-s + (−0.986 + 0.164i)6-s + (−0.879 − 0.475i)7-s + (−0.677 − 0.735i)8-s + (−0.986 − 0.164i)9-s + (0.945 + 0.324i)10-s + (−0.986 + 0.164i)11-s + (−0.401 − 0.915i)12-s + (−0.0825 + 0.996i)13-s + (0.245 − 0.969i)14-s + (0.789 + 0.614i)15-s + (0.546 − 0.837i)16-s + (−0.879 − 0.475i)17-s + ⋯
L(s)  = 1  + (0.245 + 0.969i)2-s + (−0.0825 + 0.996i)3-s + (−0.879 + 0.475i)4-s + (0.546 − 0.837i)5-s + (−0.986 + 0.164i)6-s + (−0.879 − 0.475i)7-s + (−0.677 − 0.735i)8-s + (−0.986 − 0.164i)9-s + (0.945 + 0.324i)10-s + (−0.986 + 0.164i)11-s + (−0.401 − 0.915i)12-s + (−0.0825 + 0.996i)13-s + (0.245 − 0.969i)14-s + (0.789 + 0.614i)15-s + (0.546 − 0.837i)16-s + (−0.879 − 0.475i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $0.472 - 0.881i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ 0.472 - 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1846889631 - 0.1105959234i\)
\(L(\frac12)\) \(\approx\) \(0.1846889631 - 0.1105959234i\)
\(L(1)\) \(\approx\) \(0.6005092305 + 0.3708618925i\)
\(L(1)\) \(\approx\) \(0.6005092305 + 0.3708618925i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (0.245 + 0.969i)T \)
3 \( 1 + (-0.0825 + 0.996i)T \)
5 \( 1 + (0.546 - 0.837i)T \)
7 \( 1 + (-0.879 - 0.475i)T \)
11 \( 1 + (-0.986 + 0.164i)T \)
13 \( 1 + (-0.0825 + 0.996i)T \)
17 \( 1 + (-0.879 - 0.475i)T \)
23 \( 1 + (-0.0825 - 0.996i)T \)
29 \( 1 + (-0.879 - 0.475i)T \)
31 \( 1 + (0.245 - 0.969i)T \)
37 \( 1 + (-0.986 + 0.164i)T \)
41 \( 1 + (0.789 + 0.614i)T \)
43 \( 1 + (0.945 - 0.324i)T \)
47 \( 1 + (-0.986 + 0.164i)T \)
53 \( 1 + (-0.986 + 0.164i)T \)
59 \( 1 + (0.789 + 0.614i)T \)
61 \( 1 + (-0.677 + 0.735i)T \)
67 \( 1 + (-0.677 - 0.735i)T \)
71 \( 1 + (-0.677 - 0.735i)T \)
73 \( 1 + (-0.879 - 0.475i)T \)
79 \( 1 + (0.945 - 0.324i)T \)
83 \( 1 + (0.546 + 0.837i)T \)
89 \( 1 + (-0.879 + 0.475i)T \)
97 \( 1 + (-0.677 + 0.735i)T \)
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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

−24.828466972779373135456768448942, −23.77322137878594202823365632613, −22.871231321547685033689076528863, −22.33691159412366926525154186834, −21.50003912009043925809777582335, −20.3689917908259216675879243341, −19.342150209015724787843918055324, −18.930458066859228746271088832465, −17.831860625822321635145749148, −17.639219213664304308054926340953, −15.76393748937803676346136305837, −14.75109054157002438854383786207, −13.68483041557409490285922908564, −13.03492093948383757578582568762, −12.467293940597728674607387875056, −11.17995077558787984594688059242, −10.53040176797935580805437165526, −9.47858649722837265383922566777, −8.37262333172487221602468197424, −7.1205813122656977504409109055, −5.94934030072018267312262791813, −5.38263512943937931029581306374, −3.34624189043786431691655077414, −2.72012699470443226565181578534, −1.7488288472239645320280927712, 0.11366821147225390300174414561, 2.646114228085356216017245942048, 4.121937871775074092875069881100, 4.68973395578679788482842034805, 5.75267901611350229203990796292, 6.618905826247746755150378576727, 7.95623759412383647545766061931, 9.14188525784454499947142174667, 9.52695090799692282262041008954, 10.62069799936858132439800311860, 12.11639849280621570787081160386, 13.20837132887898806952750896631, 13.7596849183004758455159410297, 14.90285039936150151872788620422, 15.945346907455203580846304596470, 16.356819407145723505653607668319, 17.07882684611752313672029514983, 17.996604546574495070677049198339, 19.27602861442084095660407201838, 20.61977261857962809504816033386, 21.04367819881702673494386874868, 22.20876272287038299916308593672, 22.70449431647689079634867111409, 23.79313790474798881683580272844, 24.49580892525177369218047182102

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