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Properties

Label 1-419-419.12-r0-0-0
Degree $1$
Conductor $419$
Sign $-0.262 + 0.964i$
Analytic cond. $1.94582$
Root an. cond. $1.94582$
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  + (0.0375 + 0.999i)2-s + (−0.816 + 0.578i)3-s + (−0.997 + 0.0750i)4-s + (−0.699 + 0.715i)5-s + (−0.608 − 0.793i)6-s + (−0.0825 − 0.996i)7-s + (−0.112 − 0.993i)8-s + (0.331 − 0.943i)9-s + (−0.740 − 0.671i)10-s + (0.977 + 0.208i)11-s + (0.770 − 0.637i)12-s + (0.415 + 0.909i)13-s + (0.992 − 0.119i)14-s + (0.157 − 0.987i)15-s + (0.988 − 0.149i)16-s + (0.807 − 0.590i)17-s + ⋯
L(s)  = 1  + (0.0375 + 0.999i)2-s + (−0.816 + 0.578i)3-s + (−0.997 + 0.0750i)4-s + (−0.699 + 0.715i)5-s + (−0.608 − 0.793i)6-s + (−0.0825 − 0.996i)7-s + (−0.112 − 0.993i)8-s + (0.331 − 0.943i)9-s + (−0.740 − 0.671i)10-s + (0.977 + 0.208i)11-s + (0.770 − 0.637i)12-s + (0.415 + 0.909i)13-s + (0.992 − 0.119i)14-s + (0.157 − 0.987i)15-s + (0.988 − 0.149i)16-s + (0.807 − 0.590i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 419 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 419 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(419\)
Sign: $-0.262 + 0.964i$
Analytic conductor: \(1.94582\)
Root analytic conductor: \(1.94582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{419} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 419,\ (0:\ ),\ -0.262 + 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4890928683 + 0.6398846194i\)
\(L(\frac12)\) \(\approx\) \(0.4890928683 + 0.6398846194i\)
\(L(1)\) \(\approx\) \(0.5868418636 + 0.4549085159i\)
\(L(1)\) \(\approx\) \(0.5868418636 + 0.4549085159i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad419 \( 1 \)
good2 \( 1 + (0.0375 + 0.999i)T \)
3 \( 1 + (-0.816 + 0.578i)T \)
5 \( 1 + (-0.699 + 0.715i)T \)
7 \( 1 + (-0.0825 - 0.996i)T \)
11 \( 1 + (0.977 + 0.208i)T \)
13 \( 1 + (0.415 + 0.909i)T \)
17 \( 1 + (0.807 - 0.590i)T \)
19 \( 1 + (-0.172 - 0.985i)T \)
23 \( 1 + (-0.455 + 0.889i)T \)
29 \( 1 + (0.303 + 0.952i)T \)
31 \( 1 + (0.274 - 0.961i)T \)
37 \( 1 + (0.620 - 0.784i)T \)
41 \( 1 + (0.387 + 0.921i)T \)
43 \( 1 + (-0.864 + 0.502i)T \)
47 \( 1 + (-0.401 - 0.915i)T \)
53 \( 1 + (-0.631 - 0.775i)T \)
59 \( 1 + (0.841 - 0.540i)T \)
61 \( 1 + (0.999 - 0.0300i)T \)
67 \( 1 + (-0.0225 + 0.999i)T \)
71 \( 1 + (0.841 + 0.540i)T \)
73 \( 1 + (0.886 + 0.462i)T \)
79 \( 1 + (0.0675 - 0.997i)T \)
83 \( 1 + (0.750 + 0.660i)T \)
89 \( 1 + (0.945 + 0.324i)T \)
97 \( 1 + (-0.345 + 0.938i)T \)
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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

−23.77592813560276180331543279556, −22.90116925919392292226749016883, −22.38147637973478112582119220602, −21.39998957385885888565168210876, −20.51528256936911957656560877118, −19.43447597251688221613887655494, −18.968083148625752104272406820492, −18.103076847302434533734313661672, −17.14569857874170393201723074805, −16.358107537360404512260506368090, −15.176029328697630736567703504143, −14.01844710254121158073764888994, −12.79547056437674574602584543836, −12.287028780761069247669348024202, −11.814304345498278657367697530903, −10.80259878775350631361802010567, −9.77667631189142850903491439450, −8.45762234512358403638447735239, −8.056151284365766885518308744510, −6.21870520519589380845133378696, −5.47789688813850036497746918906, −4.399439713995643165069459485192, −3.30080921551266292572243700458, −1.81518540161185419634967418519, −0.81989050284506760997377037280, 0.878205225589746851929776992237, 3.57470128376696834476565993041, 4.08468839071116380521906615771, 5.0754935218755897417566256353, 6.53470223358282715648396662303, 6.81673530553946028967908802073, 7.87971303041772313932063130412, 9.30204086107842275869268650233, 9.97907582963729614467954044571, 11.20211430694482958117681389996, 11.81089027229819263210302357078, 13.181649829523754358288493938, 14.30733588815428248027566323555, 14.83190142657685261332018604445, 16.029365762977134012823802683014, 16.39764481936370831433880400183, 17.35051195388070614837828631773, 18.08139098313809363566009082261, 19.11106553263944945482034910895, 20.06264951519002150326454417434, 21.472348667814560678927851732150, 22.14541040784284071258583649828, 23.05937750161498210453345285495, 23.426022051259106572473869210862, 24.1404088934853127560454087535

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