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Properties

Label 1-19e2-361.153-r0-0-0
Degree $1$
Conductor $361$
Sign $0.589 - 0.807i$
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  + (−0.986 + 0.164i)2-s + (0.546 − 0.837i)3-s + (0.945 − 0.324i)4-s + (0.789 − 0.614i)5-s + (−0.401 + 0.915i)6-s + (0.945 + 0.324i)7-s + (−0.879 + 0.475i)8-s + (−0.401 − 0.915i)9-s + (−0.677 + 0.735i)10-s + (−0.401 + 0.915i)11-s + (0.245 − 0.969i)12-s + (0.546 − 0.837i)13-s + (−0.986 − 0.164i)14-s + (−0.0825 − 0.996i)15-s + (0.789 − 0.614i)16-s + (0.945 + 0.324i)17-s + ⋯
L(s)  = 1  + (−0.986 + 0.164i)2-s + (0.546 − 0.837i)3-s + (0.945 − 0.324i)4-s + (0.789 − 0.614i)5-s + (−0.401 + 0.915i)6-s + (0.945 + 0.324i)7-s + (−0.879 + 0.475i)8-s + (−0.401 − 0.915i)9-s + (−0.677 + 0.735i)10-s + (−0.401 + 0.915i)11-s + (0.245 − 0.969i)12-s + (0.546 − 0.837i)13-s + (−0.986 − 0.164i)14-s + (−0.0825 − 0.996i)15-s + (0.789 − 0.614i)16-s + (0.945 + 0.324i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $0.589 - 0.807i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ 0.589 - 0.807i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.161183892 - 0.5897447171i\)
\(L(\frac12)\) \(\approx\) \(1.161183892 - 0.5897447171i\)
\(L(1)\) \(\approx\) \(1.014689172 - 0.2979471320i\)
\(L(1)\) \(\approx\) \(1.014689172 - 0.2979471320i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (-0.986 + 0.164i)T \)
3 \( 1 + (0.546 - 0.837i)T \)
5 \( 1 + (0.789 - 0.614i)T \)
7 \( 1 + (0.945 + 0.324i)T \)
11 \( 1 + (-0.401 + 0.915i)T \)
13 \( 1 + (0.546 - 0.837i)T \)
17 \( 1 + (0.945 + 0.324i)T \)
23 \( 1 + (0.546 + 0.837i)T \)
29 \( 1 + (0.945 + 0.324i)T \)
31 \( 1 + (-0.986 - 0.164i)T \)
37 \( 1 + (-0.401 + 0.915i)T \)
41 \( 1 + (-0.0825 - 0.996i)T \)
43 \( 1 + (-0.677 - 0.735i)T \)
47 \( 1 + (-0.401 + 0.915i)T \)
53 \( 1 + (-0.401 + 0.915i)T \)
59 \( 1 + (-0.0825 - 0.996i)T \)
61 \( 1 + (-0.879 - 0.475i)T \)
67 \( 1 + (-0.879 + 0.475i)T \)
71 \( 1 + (-0.879 + 0.475i)T \)
73 \( 1 + (0.945 + 0.324i)T \)
79 \( 1 + (-0.677 - 0.735i)T \)
83 \( 1 + (0.789 + 0.614i)T \)
89 \( 1 + (0.945 - 0.324i)T \)
97 \( 1 + (-0.879 - 0.475i)T \)
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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

−25.17197152822837861973706000728, −24.34219252096523880900843320154, −23.06444948484377201393932816271, −21.51935033332077898947491211508, −21.33930029928211835079354862138, −20.63990320613687951140534097944, −19.4912493148609314405742171268, −18.59913668909654052098862548842, −17.93459731705171347794423649305, −16.684059137672857787198518803016, −16.32881858804631787324334734352, −14.95922800983909388572938615642, −14.30582978184892843013762435016, −13.41075204611254452483501731511, −11.64525584095349931286555045727, −10.83068971884027166809245499673, −10.30783780801548765146845587533, −9.24095391916905683620672592818, −8.47297466131622134155550622739, −7.54477723873054772357234835157, −6.29447866770370619941134947160, −5.096524619542517388796038662566, −3.56740408103303743925724031047, −2.62554006637230211269372323225, −1.488113584429362571220148257859, 1.2488078413361838980330787160, 1.84028373203087460057576163089, 3.04191729461333027024111729929, 5.15909314017882491065718273799, 5.97828375389778703295211350471, 7.26113615552590868645181622128, 8.062932403134732378889883031262, 8.760862296138558836204733453412, 9.70705272508191214263868315196, 10.71468578321186215369616402743, 12.0274480031715825535460007125, 12.70667178646013762763717203936, 13.879030801256999209117136093863, 14.836146752416721656143969046177, 15.6168370017178703178775125979, 17.055405603381293999536490459851, 17.66324892428119959657720924346, 18.224263501091024430297130029066, 19.09898542265656607967885154151, 20.43927014994633057209411660837, 20.51830587962932421911526361581, 21.54859382598143880620527323539, 23.3714913562909891931919573937, 23.94919816661019715692899930109, 24.96630517479532215257614924162

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