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Properties

Label 1-837-837.437-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.892 + 0.451i$
Analytic cond. $3.88701$
Root an. cond. $3.88701$
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  + (0.882 + 0.469i)2-s + (0.559 + 0.829i)4-s + (0.939 + 0.342i)5-s + (−0.719 + 0.694i)7-s + (0.104 + 0.994i)8-s + (0.669 + 0.743i)10-s + (−0.719 + 0.694i)11-s + (−0.848 − 0.529i)13-s + (−0.961 + 0.275i)14-s + (−0.374 + 0.927i)16-s + (−0.809 + 0.587i)17-s + (−0.978 + 0.207i)19-s + (0.241 + 0.970i)20-s + (−0.961 + 0.275i)22-s + (0.961 − 0.275i)23-s + ⋯
L(s)  = 1  + (0.882 + 0.469i)2-s + (0.559 + 0.829i)4-s + (0.939 + 0.342i)5-s + (−0.719 + 0.694i)7-s + (0.104 + 0.994i)8-s + (0.669 + 0.743i)10-s + (−0.719 + 0.694i)11-s + (−0.848 − 0.529i)13-s + (−0.961 + 0.275i)14-s + (−0.374 + 0.927i)16-s + (−0.809 + 0.587i)17-s + (−0.978 + 0.207i)19-s + (0.241 + 0.970i)20-s + (−0.961 + 0.275i)22-s + (0.961 − 0.275i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.892 + 0.451i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.892 + 0.451i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4662676022 + 1.954549716i\)
\(L(\frac12)\) \(\approx\) \(0.4662676022 + 1.954549716i\)
\(L(1)\) \(\approx\) \(1.270416026 + 0.9567540377i\)
\(L(1)\) \(\approx\) \(1.270416026 + 0.9567540377i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.882 + 0.469i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (-0.719 + 0.694i)T \)
11 \( 1 + (-0.719 + 0.694i)T \)
13 \( 1 + (-0.848 - 0.529i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (0.961 - 0.275i)T \)
29 \( 1 + (-0.882 - 0.469i)T \)
37 \( 1 - T \)
41 \( 1 + (0.374 + 0.927i)T \)
43 \( 1 + (-0.0348 - 0.999i)T \)
47 \( 1 + (0.615 + 0.788i)T \)
53 \( 1 + (0.913 + 0.406i)T \)
59 \( 1 + (0.882 - 0.469i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.104 + 0.994i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.997 + 0.0697i)T \)
83 \( 1 + (0.0348 + 0.999i)T \)
89 \( 1 + (0.913 - 0.406i)T \)
97 \( 1 + (-0.241 - 0.970i)T \)
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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

−21.803213631023458163799471393004, −21.0556701313452252326611853532, −20.46118010317942889638115985724, −19.47813943422705151832961838380, −18.95871497502234687192488085238, −17.78987234845900790941878691821, −16.790649354347705201623193824482, −16.22976729289321760182284782367, −15.191721065840983973607522389225, −14.274094300128116489798587703883, −13.400093132519414478607361179544, −13.167538267379221530104152636744, −12.24712408662702378365317722940, −11.05265488079654545846545218264, −10.47790592268770941296588260255, −9.59215965089729824522439988867, −8.848820000175369522875400502403, −7.1699637279505654197921752439, −6.62361387118130202892731065737, −5.5495723880496233934327623221, −4.875957789167292319965747395940, −3.8447433544919243022206272846, −2.74429292980240234819262382076, −2.01028105116790942072596932025, −0.59028257324238003606406100761, 2.25362579876846768241649218246, 2.45561027419526027331207966084, 3.71819324257702046450144246544, 4.96248508129030545387201368404, 5.58431830216196018974092833394, 6.494472149273127721233624588185, 7.11318381494973853490580521789, 8.28146083190190705150362051392, 9.26425184524831808771498829892, 10.242455788003606813758151171748, 11.024891861467117534748360591029, 12.36664034817019653264243640949, 12.85536212999089005975369300, 13.39660016121073708441529857199, 14.613091741640251407361669132713, 15.112348430459085882280584001, 15.752522274908080246346518792568, 17.01583249513020668255054909377, 17.36252720252018480940356439699, 18.389586254649786422859948735677, 19.311212295568821633876631378504, 20.37616814977743010378002836799, 21.151002316662686235868523787, 21.83261209062011825832643245233, 22.48924567218317322319088885909

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