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Properties

Label 1-19e2-361.119-r0-0-0
Degree $1$
Conductor $361$
Sign $0.950 + 0.311i$
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  + (0.811 + 0.584i)2-s + (0.418 − 0.908i)3-s + (0.315 + 0.948i)4-s + (0.919 + 0.393i)5-s + (0.870 − 0.492i)6-s + (0.851 − 0.523i)7-s + (−0.298 + 0.954i)8-s + (−0.649 − 0.760i)9-s + (0.515 + 0.856i)10-s + (0.635 + 0.771i)11-s + (0.993 + 0.110i)12-s + (−0.263 − 0.964i)13-s + (0.997 + 0.0734i)14-s + (0.741 − 0.670i)15-s + (−0.800 + 0.599i)16-s + (−0.979 + 0.200i)17-s + ⋯
L(s)  = 1  + (0.811 + 0.584i)2-s + (0.418 − 0.908i)3-s + (0.315 + 0.948i)4-s + (0.919 + 0.393i)5-s + (0.870 − 0.492i)6-s + (0.851 − 0.523i)7-s + (−0.298 + 0.954i)8-s + (−0.649 − 0.760i)9-s + (0.515 + 0.856i)10-s + (0.635 + 0.771i)11-s + (0.993 + 0.110i)12-s + (−0.263 − 0.964i)13-s + (0.997 + 0.0734i)14-s + (0.741 − 0.670i)15-s + (−0.800 + 0.599i)16-s + (−0.979 + 0.200i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $0.950 + 0.311i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ 0.950 + 0.311i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.738787997 + 0.4379901612i\)
\(L(\frac12)\) \(\approx\) \(2.738787997 + 0.4379901612i\)
\(L(1)\) \(\approx\) \(2.095679255 + 0.2750885435i\)
\(L(1)\) \(\approx\) \(2.095679255 + 0.2750885435i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (0.811 + 0.584i)T \)
3 \( 1 + (0.418 - 0.908i)T \)
5 \( 1 + (0.919 + 0.393i)T \)
7 \( 1 + (0.851 - 0.523i)T \)
11 \( 1 + (0.635 + 0.771i)T \)
13 \( 1 + (-0.263 - 0.964i)T \)
17 \( 1 + (-0.979 + 0.200i)T \)
23 \( 1 + (-0.995 - 0.0917i)T \)
29 \( 1 + (-0.367 - 0.929i)T \)
31 \( 1 + (0.716 + 0.697i)T \)
37 \( 1 + (-0.986 + 0.164i)T \)
41 \( 1 + (-0.531 - 0.847i)T \)
43 \( 1 + (0.484 + 0.875i)T \)
47 \( 1 + (-0.861 - 0.507i)T \)
53 \( 1 + (-0.00918 + 0.999i)T \)
59 \( 1 + (0.999 - 0.0367i)T \)
61 \( 1 + (0.606 + 0.794i)T \)
67 \( 1 + (-0.0459 - 0.998i)T \)
71 \( 1 + (0.606 - 0.794i)T \)
73 \( 1 + (0.663 + 0.748i)T \)
79 \( 1 + (-0.999 - 0.0183i)T \)
83 \( 1 + (-0.998 + 0.0550i)T \)
89 \( 1 + (0.663 - 0.748i)T \)
97 \( 1 + (-0.0459 + 0.998i)T \)
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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

−24.428858499947136448119382310530, −24.076441174077292121921186342765, −22.33057745787279322066307937727, −21.94492988260694847156319587669, −21.25351455262695087534416163591, −20.610548998453900193420894907629, −19.74007379288291293849654130671, −18.736498064488690748529429457, −17.53898404371715590043153922127, −16.45464424389940187768345152076, −15.59125997835727432265815907369, −14.37376503723899588248138110177, −14.16317593473072503869008348002, −13.16314143596466312349837199588, −11.78624996016885238753102429130, −11.19706678593082243181833829907, −10.07393738460722542867805548621, −9.210007258610991420954767430489, −8.51267693838615503833412701806, −6.56338242172173451429748406706, −5.5192174770560144509066920059, −4.758067422309578542883702190605, −3.84871704844015263419304670928, −2.46366266285856811704581970727, −1.70132509370170002743487698460, 1.70353509596212186571925718458, 2.53298998682820449126957548642, 3.87890376790581203415753649258, 5.11152301345105380320863829518, 6.22262080078672128284907591307, 6.97179395200236320910396461934, 7.831011953429033903669688089116, 8.780299983196599504480263064073, 10.1897880801668583639743784112, 11.4510849422361251465946562415, 12.402016751542245793640059559819, 13.321024213729007554879872593959, 13.98653932667953004210409209689, 14.661951158076830910125019915175, 15.43855852943767622899550684960, 17.195952012430767611853730197844, 17.505991824848207716105073794041, 18.1730772664519182415735943647, 19.760168643615490498372045229245, 20.45010193120713449094242777921, 21.28454292849140024726783417471, 22.46318795628413065952967615213, 22.943244429825048315028262207666, 24.251707221648978817022644981335, 24.55668285938525355632761668220

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