Properties

Label 1-19-19.16-r0-0-0
Degree $1$
Conductor $19$
Sign $0.756 + 0.654i$
Analytic cond. $0.0882356$
Root an. cond. $0.0882356$
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  + (0.173 + 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.939 − 0.342i)5-s + (0.766 + 0.642i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (0.173 − 0.984i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.766 + 0.642i)13-s + (−0.939 − 0.342i)14-s + (−0.939 + 0.342i)15-s + (0.766 − 0.642i)16-s + (0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.939 − 0.342i)5-s + (0.766 + 0.642i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (0.173 − 0.984i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.766 + 0.642i)13-s + (−0.939 − 0.342i)14-s + (−0.939 + 0.342i)15-s + (0.766 − 0.642i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(19\)
Sign: $0.756 + 0.654i$
Analytic conductor: \(0.0882356\)
Root analytic conductor: \(0.0882356\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 19,\ (0:\ ),\ 0.756 + 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6490920513 + 0.2418349987i\)
\(L(\frac12)\) \(\approx\) \(0.6490920513 + 0.2418349987i\)
\(L(1)\) \(\approx\) \(0.9148777962 + 0.2743962000i\)
\(L(1)\) \(\approx\) \(0.9148777962 + 0.2743962000i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (0.173 + 0.984i)T \)
3 \( 1 + (0.766 - 0.642i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.766 + 0.642i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (0.173 - 0.984i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (-0.939 + 0.342i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (0.173 + 0.984i)T \)
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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

−39.966678468297088389924664394, −38.79794059860006376489807231668, −38.25102936823066918362673207345, −36.69033350196856596357589369148, −35.65457837287952981220758062472, −33.3602343152603013879218081086, −32.074025094767684882108684988209, −31.02200993497157192939617987746, −29.97308117879326510240171457494, −28.0860176070842113378064833370, −27.01930258664812602884351785427, −25.90103103262920986888298480333, −23.38903181380005856896111069424, −22.37992520140482266628026460011, −20.500211947805143210805262350228, −19.95633891187922612740804620497, −18.42318407229489172366645497977, −15.97183270410677329366944613353, −14.45929740897332565709300613413, −12.961483732421802065524330534898, −10.97732081444656069231364161663, −9.784705571944299027829113931934, −7.87046245161347143463711975430, −4.40326503236999735608592608485, −3.098780796516242989029474866290, 3.63715141026986389406139395986, 6.144738739176663757367474316151, 7.96383872105274116227731779107, 8.919819492265205268438072345589, 12.219233258408520928330202026637, 13.5235662208293974187418359784, 15.15475284576543290999489815175, 16.231741681749722530869331618, 18.399608549824534518277132803751, 19.364707078271979697428408939988, 21.45392796599038916835575109704, 23.40258719388946822596410683783, 24.262187876552078686309678741301, 25.58986480145456883594150043559, 26.683928658170236982917790133, 28.38185295028579843677332387969, 30.57937888475565841125168579009, 31.6008207000210932899107105769, 32.348634645340665716708419904855, 34.48649083564432440906150833866, 35.3379868267910960637953750558, 36.33581719795455556071506717654, 37.92800505845793264133536334229, 39.75253373678277456012646920810, 41.20384797788594438920615324210

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