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Properties

Label 1-1832-1832.189-r1-0-0
Degree $1$
Conductor $1832$
Sign $0.110 - 0.993i$
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  + (−0.716 + 0.697i)3-s + (−0.298 − 0.954i)5-s + (0.936 + 0.350i)7-s + (0.0275 − 0.999i)9-s + (0.789 − 0.614i)11-s + (−0.735 − 0.677i)13-s + (0.879 + 0.475i)15-s + (−0.677 + 0.735i)17-s + (−0.975 + 0.218i)19-s + (−0.915 + 0.401i)21-s + (−0.981 + 0.191i)23-s + (−0.821 + 0.569i)25-s + (0.677 + 0.735i)27-s + (−0.771 + 0.635i)29-s + (0.990 − 0.137i)31-s + ⋯
L(s)  = 1  + (−0.716 + 0.697i)3-s + (−0.298 − 0.954i)5-s + (0.936 + 0.350i)7-s + (0.0275 − 0.999i)9-s + (0.789 − 0.614i)11-s + (−0.735 − 0.677i)13-s + (0.879 + 0.475i)15-s + (−0.677 + 0.735i)17-s + (−0.975 + 0.218i)19-s + (−0.915 + 0.401i)21-s + (−0.981 + 0.191i)23-s + (−0.821 + 0.569i)25-s + (0.677 + 0.735i)27-s + (−0.771 + 0.635i)29-s + (0.990 − 0.137i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1832\)    =    \(2^{3} \cdot 229\)
Sign: $0.110 - 0.993i$
Analytic conductor: \(196.875\)
Root analytic conductor: \(196.875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1832} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1832,\ (1:\ ),\ 0.110 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8181084313 - 0.7322194261i\)
\(L(\frac12)\) \(\approx\) \(0.8181084313 - 0.7322194261i\)
\(L(1)\) \(\approx\) \(0.8218710156 + 0.02362888602i\)
\(L(1)\) \(\approx\) \(0.8218710156 + 0.02362888602i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
229 \( 1 \)
good3 \( 1 + (-0.716 + 0.697i)T \)
5 \( 1 + (-0.298 - 0.954i)T \)
7 \( 1 + (0.936 + 0.350i)T \)
11 \( 1 + (0.789 - 0.614i)T \)
13 \( 1 + (-0.735 - 0.677i)T \)
17 \( 1 + (-0.677 + 0.735i)T \)
19 \( 1 + (-0.975 + 0.218i)T \)
23 \( 1 + (-0.981 + 0.191i)T \)
29 \( 1 + (-0.771 + 0.635i)T \)
31 \( 1 + (0.990 - 0.137i)T \)
37 \( 1 + (0.592 - 0.805i)T \)
41 \( 1 + (0.523 + 0.851i)T \)
43 \( 1 + (0.401 + 0.915i)T \)
47 \( 1 + (0.999 + 0.0275i)T \)
53 \( 1 + (-0.245 - 0.969i)T \)
59 \( 1 + (0.805 - 0.592i)T \)
61 \( 1 + (0.879 - 0.475i)T \)
67 \( 1 + (0.523 - 0.851i)T \)
71 \( 1 + (0.926 - 0.376i)T \)
73 \( 1 + (0.569 + 0.821i)T \)
79 \( 1 + (0.771 + 0.635i)T \)
83 \( 1 + (-0.993 + 0.110i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (-0.904 + 0.426i)T \)
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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

−19.96087425553358475757182319186, −19.261329232511770750631883269568, −18.65470237464659480672032240335, −17.84679752204812733294120956741, −17.34266782434602536852667394302, −16.78944990663086997610181033059, −15.63839142770636127055205546622, −14.91362343973451014479216227490, −14.09490624085153281931437680746, −13.70247616352041393679872255818, −12.46212973524734904704984868571, −11.77522199309224332180015506046, −11.37279377854997751753633549021, −10.57861054933734934500038583011, −9.81403006816502497608506438960, −8.63954192187996760356366431222, −7.69374636427103495965819873103, −7.08612643332987050337362699582, −6.60203175120194589233689929929, −5.65564551259623327953699912331, −4.41995568513242970852692261563, −4.19094907912053097499497743630, −2.364693378092052919337283175272, −2.0916527785032568707136330665, −0.78032345406003977167294108984, 0.29361142044757109671513423819, 1.21517169770843091947419230783, 2.28906740564297056333724744282, 3.809986821669432814436576277781, 4.26262803816041798335488814471, 5.10966161424386767537431148438, 5.76387441578344796236244026174, 6.542029964032646059711402438708, 7.9167466167868541779037780306, 8.43105215410562379461079249404, 9.25242913791985638691676003461, 9.98482828526058384659753650439, 11.17196196525981956935782988143, 11.300648473973960733807117043998, 12.46470230781234721846230819000, 12.669002062269305737948247986680, 14.04609653683795164439237300720, 14.850141997619731229767251105259, 15.37667083028618326069849791980, 16.23279443489707181661944906479, 16.90492797040151406814141049463, 17.45244402276387006303981969811, 18.011502491947101020712095543503, 19.22927475424991534770242811363, 19.90252514855810285966206307496

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