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Properties

Label 1-1832-1832.1355-r1-0-0
Degree $1$
Conductor $1832$
Sign $0.176 + 0.984i$
Analytic cond. $196.875$
Root an. cond. $196.875$
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  + (0.904 − 0.426i)3-s + (0.962 + 0.272i)5-s + (0.451 + 0.892i)7-s + (0.635 − 0.771i)9-s + (−0.677 + 0.735i)11-s + (0.245 + 0.969i)13-s + (0.986 − 0.164i)15-s + (0.245 + 0.969i)17-s + (0.716 + 0.697i)19-s + (0.789 + 0.614i)21-s + (0.993 − 0.110i)23-s + (0.851 + 0.523i)25-s + (0.245 − 0.969i)27-s + (−0.998 − 0.0550i)29-s + (−0.298 − 0.954i)31-s + ⋯
L(s)  = 1  + (0.904 − 0.426i)3-s + (0.962 + 0.272i)5-s + (0.451 + 0.892i)7-s + (0.635 − 0.771i)9-s + (−0.677 + 0.735i)11-s + (0.245 + 0.969i)13-s + (0.986 − 0.164i)15-s + (0.245 + 0.969i)17-s + (0.716 + 0.697i)19-s + (0.789 + 0.614i)21-s + (0.993 − 0.110i)23-s + (0.851 + 0.523i)25-s + (0.245 − 0.969i)27-s + (−0.998 − 0.0550i)29-s + (−0.298 − 0.954i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1832\)    =    \(2^{3} \cdot 229\)
Sign: $0.176 + 0.984i$
Analytic conductor: \(196.875\)
Root analytic conductor: \(196.875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1832} (1355, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1832,\ (1:\ ),\ 0.176 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.368021362 + 2.816683052i\)
\(L(\frac12)\) \(\approx\) \(3.368021362 + 2.816683052i\)
\(L(1)\) \(\approx\) \(1.831426475 + 0.3958742845i\)
\(L(1)\) \(\approx\) \(1.831426475 + 0.3958742845i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
229 \( 1 \)
good3 \( 1 + (0.904 - 0.426i)T \)
5 \( 1 + (0.962 + 0.272i)T \)
7 \( 1 + (0.451 + 0.892i)T \)
11 \( 1 + (-0.677 + 0.735i)T \)
13 \( 1 + (0.245 + 0.969i)T \)
17 \( 1 + (0.245 + 0.969i)T \)
19 \( 1 + (0.716 + 0.697i)T \)
23 \( 1 + (0.993 - 0.110i)T \)
29 \( 1 + (-0.998 - 0.0550i)T \)
31 \( 1 + (-0.298 - 0.954i)T \)
37 \( 1 + (-0.137 + 0.990i)T \)
41 \( 1 + (-0.350 + 0.936i)T \)
43 \( 1 + (0.789 + 0.614i)T \)
47 \( 1 + (0.635 - 0.771i)T \)
53 \( 1 + (0.0825 - 0.996i)T \)
59 \( 1 + (-0.137 - 0.990i)T \)
61 \( 1 + (0.986 + 0.164i)T \)
67 \( 1 + (-0.350 - 0.936i)T \)
71 \( 1 + (-0.975 + 0.218i)T \)
73 \( 1 + (-0.851 - 0.523i)T \)
79 \( 1 + (-0.998 + 0.0550i)T \)
83 \( 1 + (-0.926 - 0.376i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.0275 + 0.999i)T \)
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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

−20.09364325111880184508881695773, −19.134048183011029000179480916739, −18.29989304548826585945084151437, −17.645094918749546515029991668221, −16.83336632618965379045435991881, −16.046526071102147855931057154478, −15.46878087359442160674665189736, −14.32838650097190201694439480689, −13.98513815944358559597434461537, −13.25745160882214489693070014896, −12.80340964824326134513880385207, −11.30105677312609119479371814712, −10.58140772454306565386510747556, −10.12760852072344195699434938513, −9.04565515589187419532038666476, −8.736499506209880191689611626411, −7.45499681883613212576338551963, −7.26508339265661177917427072066, −5.57688290659484733369630397833, −5.25158068695271979898625453422, −4.23677861822887400541390662639, −3.15109815336524077784448843485, −2.65791052143330328583580458003, −1.42647269588002134038517723390, −0.62400095686096841840293741047, 1.38899293976491241529534994953, 1.90342024593765496627069019527, 2.61691735729265431839976126491, 3.54830785083156216441623847019, 4.681711298758590008701911691357, 5.6158370984750470741663017243, 6.36340174220003854045103438982, 7.25231415168683225079355750030, 8.03259088885204130605756321369, 8.84088140949792163589573335784, 9.52245309691481482144460423082, 10.10509533862199921653024532954, 11.20780956675932941874941116559, 12.072451842337015368222418665, 12.97283519453206264621773591100, 13.3341339189552764203061283601, 14.42137151011602930887735636041, 14.75778522207143537187822016732, 15.410541916894250924833998568894, 16.509456219730742023646397629157, 17.396944473460289271786605941252, 18.136005979352680217877122999657, 18.75055586906522070834548457211, 19.045577685841931218197633347533, 20.42897161486810714544360365887

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