L-function 1-1700-1700.1391-r0-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
	17	$1$
good	3	$1 + (-0.972 + 0.233i)T$
	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	$1 + (-0.649 + 0.760i)T$

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