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Properties

Label 1-19e2-361.140-r0-0-0
Degree $1$
Conductor $361$
Sign $-0.999 - 0.0319i$
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  + (0.716 + 0.697i)2-s + (0.904 + 0.426i)3-s + (0.0275 + 0.999i)4-s + (−0.998 + 0.0550i)5-s + (0.350 + 0.936i)6-s + (−0.879 + 0.475i)7-s + (−0.677 + 0.735i)8-s + (0.635 + 0.771i)9-s + (−0.754 − 0.656i)10-s + (−0.986 − 0.164i)11-s + (−0.401 + 0.915i)12-s + (−0.821 + 0.569i)13-s + (−0.962 − 0.272i)14-s + (−0.926 − 0.376i)15-s + (−0.998 + 0.0550i)16-s + (0.0275 − 0.999i)17-s + ⋯
L(s)  = 1  + (0.716 + 0.697i)2-s + (0.904 + 0.426i)3-s + (0.0275 + 0.999i)4-s + (−0.998 + 0.0550i)5-s + (0.350 + 0.936i)6-s + (−0.879 + 0.475i)7-s + (−0.677 + 0.735i)8-s + (0.635 + 0.771i)9-s + (−0.754 − 0.656i)10-s + (−0.986 − 0.164i)11-s + (−0.401 + 0.915i)12-s + (−0.821 + 0.569i)13-s + (−0.962 − 0.272i)14-s + (−0.926 − 0.376i)15-s + (−0.998 + 0.0550i)16-s + (0.0275 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $-0.999 - 0.0319i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ -0.999 - 0.0319i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02238082698 + 1.402668915i\)
\(L(\frac12)\) \(\approx\) \(0.02238082698 + 1.402668915i\)
\(L(1)\) \(\approx\) \(0.8977855609 + 0.9751305029i\)
\(L(1)\) \(\approx\) \(0.8977855609 + 0.9751305029i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (0.716 + 0.697i)T \)
3 \( 1 + (0.904 + 0.426i)T \)
5 \( 1 + (-0.998 + 0.0550i)T \)
7 \( 1 + (-0.879 + 0.475i)T \)
11 \( 1 + (-0.986 - 0.164i)T \)
13 \( 1 + (-0.821 + 0.569i)T \)
17 \( 1 + (0.0275 - 0.999i)T \)
23 \( 1 + (0.904 - 0.426i)T \)
29 \( 1 + (0.851 + 0.523i)T \)
31 \( 1 + (0.245 + 0.969i)T \)
37 \( 1 + (-0.986 - 0.164i)T \)
41 \( 1 + (0.137 + 0.990i)T \)
43 \( 1 + (-0.754 + 0.656i)T \)
47 \( 1 + (0.350 + 0.936i)T \)
53 \( 1 + (0.350 + 0.936i)T \)
59 \( 1 + (0.137 + 0.990i)T \)
61 \( 1 + (-0.298 + 0.954i)T \)
67 \( 1 + (0.975 + 0.218i)T \)
71 \( 1 + (-0.298 - 0.954i)T \)
73 \( 1 + (0.0275 - 0.999i)T \)
79 \( 1 + (-0.754 + 0.656i)T \)
83 \( 1 + (0.546 - 0.837i)T \)
89 \( 1 + (0.0275 + 0.999i)T \)
97 \( 1 + (0.975 - 0.218i)T \)
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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

−24.0324836554248778019748059151, −23.38866274645000859230491304918, −22.69023712048234429025835483739, −21.52542057663801427432016119615, −20.52547841732569553186848229134, −19.94125614826903803167274129094, −19.180333069119563841348463327326, −18.75264957883336163486006982347, −17.24796858848655181724964635738, −15.572685337133337148217207135178, −15.39570189230567593008457339769, −14.280297827353693735274089968155, −13.150836085068905858204659961427, −12.775819548893887116564468510225, −11.87418715133268132264684896745, −10.51452866193657719412196231789, −9.8688443739075724081454009689, −8.56437303191274532254487698360, −7.50491048688031022362769661380, −6.65100472839086874277659986288, −5.146520841713321054660411845319, −3.92046583830826574059593368557, −3.24004288719586976900679861015, −2.26374122642859674973730436013, −0.57656895720889431497231246673, 2.80115163176448730419399789094, 3.05915475047628042793154679535, 4.45362521541523951694610853019, 5.12748143805488936939868055895, 6.78104203684007617566754529597, 7.46524663828920652174615087772, 8.47896772532934179410647983712, 9.25274556274936579289749477181, 10.57677229429633710340114595803, 11.928491633064657551941032269674, 12.69446098830594350300369276878, 13.63373319122350897432665599104, 14.59196615887993504562480097729, 15.366997708770670298189966570297, 16.04755560815705138622831322085, 16.528437909694466382675504428648, 18.22620177601034559276876845321, 19.1388071768612654190367700645, 19.94011795808733627529156479217, 20.96675750120549211913926039316, 21.70786233638499219669781783868, 22.647927984318137775551514415036, 23.3588784442156494576647453879, 24.418642279144452599262502192107, 25.08143479530037517458508505706

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