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Properties

Label 1-1700-1700.1391-r0-0-0
Degree $1$
Conductor $1700$
Sign $0.740 - 0.671i$
Analytic cond. $7.89476$
Root an. cond. $7.89476$
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 + (−0.649 − 0.760i)11-s + (0.951 + 0.309i)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 + (−0.233 − 0.972i)29-s + (0.522 + 0.852i)31-s + (0.809 + 0.587i)33-s + (−0.649 + 0.760i)37-s + (−0.996 − 0.0784i)39-s + (0.996 − 0.0784i)41-s + ⋯
L(s)  = 1  + (−0.972 + 0.233i)3-s + (−0.382 − 0.923i)7-s + (0.891 − 0.453i)9-s + (−0.649 − 0.760i)11-s + (0.951 + 0.309i)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 + (−0.233 − 0.972i)29-s + (0.522 + 0.852i)31-s + (0.809 + 0.587i)33-s + (−0.649 + 0.760i)37-s + (−0.996 − 0.0784i)39-s + (0.996 − 0.0784i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $0.740 - 0.671i$
Analytic conductor: \(7.89476\)
Root analytic conductor: \(7.89476\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (1391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1700,\ (0:\ ),\ 0.740 - 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9892108230 - 0.3817505143i\)
\(L(\frac12)\) \(\approx\) \(0.9892108230 - 0.3817505143i\)
\(L(1)\) \(\approx\) \(0.8114174476 - 0.08950248029i\)
\(L(1)\) \(\approx\) \(0.8114174476 - 0.08950248029i\)

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 \( 1 + (-0.649 - 0.760i)T \)
13 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (0.987 - 0.156i)T \)
23 \( 1 + (0.649 + 0.760i)T \)
29 \( 1 + (-0.233 - 0.972i)T \)
31 \( 1 + (0.522 + 0.852i)T \)
37 \( 1 + (-0.649 + 0.760i)T \)
41 \( 1 + (0.996 - 0.0784i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (0.587 + 0.809i)T \)
53 \( 1 + (-0.156 + 0.987i)T \)
59 \( 1 + (0.891 - 0.453i)T \)
61 \( 1 + (0.760 - 0.649i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (-0.972 + 0.233i)T \)
73 \( 1 + (0.996 + 0.0784i)T \)
79 \( 1 + (0.522 - 0.852i)T \)
83 \( 1 + (-0.987 + 0.156i)T \)
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

−20.637071161054479171988963958050, −19.50404870555162454212433158893, −18.69299152252920814520305345875, −18.12318383613684634264705605823, −17.74347792769295704353825635410, −16.56623825247794786476328192070, −16.051909689183877828579662710025, −15.41807875340019450746954682533, −14.606325772380390618908662400930, −13.317175355316247135599896944248, −12.90167521467628034168455678782, −12.14932217750985750503291793002, −11.46978822093038021617921330432, −10.63760087815582981738442627027, −9.94432683019214262398006153301, −9.056704022458572325184483346815, −8.117120577114200720703422235300, −7.20592294595407168661741885562, −6.47206769896714553342338838870, −5.563637217088214479594864320777, −5.16389069358704106679970807378, −4.04696299975488648884579499896, −2.91487787633447502074323405590, −1.972803529012382556888293859731, −0.854358982855034274947511546159, 0.63438489289380589003018459895, 1.422548951380215449954448138364, 3.07382180643861192131659982875, 3.75085436901541900589421368278, 4.68674248068302143277730620851, 5.539647693732350332761844071965, 6.25968603278647302510529064135, 7.06168009302515837842530216897, 7.82173240826014397426813971320, 8.93903148657543202098687040022, 9.81295445353197849215608680095, 10.52899647655061692650336417023, 11.15506977343001032514998863918, 11.7607508359061579722634653307, 12.80880850699009361427008096024, 13.52950442318282707837465075805, 13.993279155386098042697210421360, 15.44728070760333327334469698473, 15.867142934037043735336089894988, 16.51956052380499917014363421822, 17.25409110395752091307601288114, 17.886997147461657907741826168084, 18.77471272247380483123383729554, 19.30283270799059229273834187536, 20.50881986877819711311839294070

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