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Properties

Label 1-19e2-361.39-r0-0-0
Degree $1$
Conductor $361$
Sign $-0.151 - 0.988i$
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  + (−0.677 + 0.735i)2-s + (0.245 + 0.969i)3-s + (−0.0825 − 0.996i)4-s + (−0.986 + 0.164i)5-s + (−0.879 − 0.475i)6-s + (−0.0825 + 0.996i)7-s + (0.789 + 0.614i)8-s + (−0.879 + 0.475i)9-s + (0.546 − 0.837i)10-s + (−0.879 − 0.475i)11-s + (0.945 − 0.324i)12-s + (0.245 + 0.969i)13-s + (−0.677 − 0.735i)14-s + (−0.401 − 0.915i)15-s + (−0.986 + 0.164i)16-s + (−0.0825 + 0.996i)17-s + ⋯
L(s)  = 1  + (−0.677 + 0.735i)2-s + (0.245 + 0.969i)3-s + (−0.0825 − 0.996i)4-s + (−0.986 + 0.164i)5-s + (−0.879 − 0.475i)6-s + (−0.0825 + 0.996i)7-s + (0.789 + 0.614i)8-s + (−0.879 + 0.475i)9-s + (0.546 − 0.837i)10-s + (−0.879 − 0.475i)11-s + (0.945 − 0.324i)12-s + (0.245 + 0.969i)13-s + (−0.677 − 0.735i)14-s + (−0.401 − 0.915i)15-s + (−0.986 + 0.164i)16-s + (−0.0825 + 0.996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $-0.151 - 0.988i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ -0.151 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1725354034 + 0.2010367762i\)
\(L(\frac12)\) \(\approx\) \(-0.1725354034 + 0.2010367762i\)
\(L(1)\) \(\approx\) \(0.3450031761 + 0.3945238818i\)
\(L(1)\) \(\approx\) \(0.3450031761 + 0.3945238818i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (-0.677 + 0.735i)T \)
3 \( 1 + (0.245 + 0.969i)T \)
5 \( 1 + (-0.986 + 0.164i)T \)
7 \( 1 + (-0.0825 + 0.996i)T \)
11 \( 1 + (-0.879 - 0.475i)T \)
13 \( 1 + (0.245 + 0.969i)T \)
17 \( 1 + (-0.0825 + 0.996i)T \)
23 \( 1 + (0.245 - 0.969i)T \)
29 \( 1 + (-0.0825 + 0.996i)T \)
31 \( 1 + (-0.677 - 0.735i)T \)
37 \( 1 + (-0.879 - 0.475i)T \)
41 \( 1 + (-0.401 - 0.915i)T \)
43 \( 1 + (0.546 + 0.837i)T \)
47 \( 1 + (-0.879 - 0.475i)T \)
53 \( 1 + (-0.879 - 0.475i)T \)
59 \( 1 + (-0.401 - 0.915i)T \)
61 \( 1 + (0.789 - 0.614i)T \)
67 \( 1 + (0.789 + 0.614i)T \)
71 \( 1 + (0.789 + 0.614i)T \)
73 \( 1 + (-0.0825 + 0.996i)T \)
79 \( 1 + (0.546 + 0.837i)T \)
83 \( 1 + (-0.986 - 0.164i)T \)
89 \( 1 + (-0.0825 - 0.996i)T \)
97 \( 1 + (0.789 - 0.614i)T \)
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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

−24.009535709941718570374706383775, −23.11383816893230772743673276181, −22.6633317301753969562556439592, −20.88211467961579272175847315777, −20.31962045164623616212710354420, −19.71949660367992572222524012760, −18.90077014111163008829713074946, −18.0172915585986726049598321963, −17.344580295722822160493335987470, −16.22687598434341762446442224750, −15.280681197721375022192139060003, −13.7414274804591143926290649791, −13.08131785002946479065946491388, −12.26680325815702162641598781569, −11.35406538803924635379796334386, −10.5246013396846940884733162462, −9.33864669250250937773259831576, −8.08865326175719987596491772552, −7.65197175836765467430826082781, −6.90460402072349638513130240085, −4.99164048645339751788945545096, −3.61099024077266526681350918759, −2.831550137038038371146608326673, −1.33861694679527910888695912373, −0.20513797073748579367196953482, 2.203294605349059529411110437572, 3.5750384816118504057255920463, 4.76470221451596856542059280325, 5.68797078358634501072550011070, 6.84593131303578736435274304309, 8.27755627712234451238156765954, 8.55234273458704212993420838051, 9.59943973681021526164315502933, 10.78688253446126405084043037415, 11.28518444539691430470315027248, 12.73118012778862849177438023551, 14.300294963461573008758876592116, 14.86038347612670302415023236177, 15.85426915868766017952342124613, 16.07294215337460965743336648786, 17.129228180635783126678406142755, 18.548041345482441242179504048221, 18.95870099145126864653667813543, 19.89763318676535282102703184660, 20.93539710725837837055558908142, 21.91000729516777022461412580505, 22.8180235809066878615427761031, 23.79901502958710295362270542484, 24.45683123654581776904100553137, 25.775184585708867724247997397808

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