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Properties

Label 1-1700-1700.1287-r1-0-0
Degree $1$
Conductor $1700$
Sign $-0.732 + 0.680i$
Analytic cond. $182.690$
Root an. cond. $182.690$
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  + (−0.972 + 0.233i)3-s + (−0.382 − 0.923i)7-s + (0.891 − 0.453i)9-s + (−0.760 + 0.649i)11-s + (0.309 − 0.951i)13-s + (0.987 − 0.156i)19-s + (0.587 + 0.809i)21-s + (−0.649 − 0.760i)23-s + (−0.760 + 0.649i)27-s + (−0.972 + 0.233i)29-s + (−0.852 + 0.522i)31-s + (0.587 − 0.809i)33-s + (0.649 − 0.760i)37-s + (−0.0784 + 0.996i)39-s + (−0.0784 − 0.996i)41-s + ⋯
L(s)  = 1  + (−0.972 + 0.233i)3-s + (−0.382 − 0.923i)7-s + (0.891 − 0.453i)9-s + (−0.760 + 0.649i)11-s + (0.309 − 0.951i)13-s + (0.987 − 0.156i)19-s + (0.587 + 0.809i)21-s + (−0.649 − 0.760i)23-s + (−0.760 + 0.649i)27-s + (−0.972 + 0.233i)29-s + (−0.852 + 0.522i)31-s + (0.587 − 0.809i)33-s + (0.649 − 0.760i)37-s + (−0.0784 + 0.996i)39-s + (−0.0784 − 0.996i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $-0.732 + 0.680i$
Analytic conductor: \(182.690\)
Root analytic conductor: \(182.690\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (1287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1700,\ (1:\ ),\ -0.732 + 0.680i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05713930315 - 0.1453521069i\)
\(L(\frac12)\) \(\approx\) \(-0.05713930315 - 0.1453521069i\)
\(L(1)\) \(\approx\) \(0.6405246315 - 0.1146018336i\)
\(L(1)\) \(\approx\) \(0.6405246315 - 0.1146018336i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good3 \( 1 + (-0.972 + 0.233i)T \)
7 \( 1 + (-0.382 - 0.923i)T \)
11 \( 1 + (-0.760 + 0.649i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.987 - 0.156i)T \)
23 \( 1 + (-0.649 - 0.760i)T \)
29 \( 1 + (-0.972 + 0.233i)T \)
31 \( 1 + (-0.852 + 0.522i)T \)
37 \( 1 + (0.649 - 0.760i)T \)
41 \( 1 + (-0.0784 - 0.996i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (-0.987 - 0.156i)T \)
59 \( 1 + (0.891 - 0.453i)T \)
61 \( 1 + (-0.649 - 0.760i)T \)
67 \( 1 + (0.587 - 0.809i)T \)
71 \( 1 + (-0.233 - 0.972i)T \)
73 \( 1 + (0.996 + 0.0784i)T \)
79 \( 1 + (0.852 + 0.522i)T \)
83 \( 1 + (-0.156 - 0.987i)T \)
89 \( 1 + (-0.951 + 0.309i)T \)
97 \( 1 + (0.233 + 0.972i)T \)
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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

−20.81138507766362923533495385441, −19.67764571449104537298168712630, −18.83070553559557175406451978451, −18.411419971323708585872831497930, −17.857010171610723799795042977478, −16.6939690297686093171035338910, −16.25273427194992148036807793407, −15.67250778264199328739890281246, −14.75230311679075732611582416096, −13.610248235858582549312689235403, −13.113970161994280911062135981536, −12.258571325804053215516517486311, −11.40761402716776039126582232004, −11.20757070486269647526489854227, −9.870246528016972470496592148303, −9.447999971939261799823708801681, −8.28069555835350538652463346340, −7.52315529717088265918784638693, −6.55593752060550365921676335694, −5.78204564955996429231778399837, −5.37664644715521709126477559202, −4.272507869382365215986002657023, −3.23900728827648467745325887188, −2.16426125695886391382825872515, −1.22787211933943854582374942140, 0.04788870806998440984402674490, 0.70948687532335482633525572127, 1.898968920273273695850604841496, 3.256717979989339435079031111485, 3.99619503797288052037060191752, 4.97590374396488621498785084429, 5.58348951736799764863437446815, 6.51999194648010333215427249792, 7.356665385631898209070315876633, 7.89355115276659322356592533378, 9.33067049135354788913926263828, 9.946768624456171789188303515316, 10.75183010732425604364994237471, 11.06410833568658862650814837857, 12.408349227536048872897307497505, 12.697552814065934334805473146269, 13.56772912709438617870034895873, 14.50770829478864379116605280409, 15.51099785706110892116529111485, 16.03631443426777812734530923532, 16.666668488061805484217779698313, 17.57208967159912898040551210412, 18.01630233049642994156920301698, 18.71937184616377066963764597591, 19.905898322625713759568027190336

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