L-function 1-85-85.23-r0-0-0

• Label: 1-85-85.23-r0-0-0
• Degree: $1$
• Conductor: $85$
• Sign: $0.940 - 0.340i$
• Analytic cond.: $0.394738$
• Root an. cond.: $0.394738$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.707 − 0.707i)2^-s + (0.382 + 0.923i)3^-s − i·4^-s + (0.923 + 0.382i)6^-s + (0.923 + 0.382i)7^-s + (−0.707 − 0.707i)8^-s + (−0.707 + 0.707i)9^-s + (−0.923 − 0.382i)11^-s + (0.923 − 0.382i)12^-s + 13^-s + (0.923 − 0.382i)14^-s − 16^-s + i·18^-s + (−0.707 − 0.707i)19^-s + i·21^-s + (−0.923 + 0.382i)22^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$85$    =    $5 \cdot 17$
Sign:	$0.940 - 0.340i$
Analytic conductor:	$0.394738$
Root analytic conductor:	$0.394738$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{85} (23, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 85,\ (0:\ ),\ 0.940 - 0.340i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.501091477 - 0.2631715642i$
$L(1)$	$\approx$	$1.525710224 - 0.2290352218i$

Euler product

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

	$p$	$F_p(T)$
bad	5	$1$
	17	$1$
good	2	$1 + (0.707 - 0.707i)T$
	3	$1 + (0.382 + 0.923i)T$
	7	$1 + (0.923 + 0.382i)T$
	11	$1 + (-0.923 - 0.382i)T$
	13	$1 + T$
	19	$1 + (-0.707 - 0.707i)T$
	23	$1 + (-0.382 + 0.923i)T$
	29	$1 + (-0.382 - 0.923i)T$
	31	$1 + (-0.923 + 0.382i)T$

Imaginary part of the first few zeros on the critical line

−30.87712422139981104873630815072, −30.19473093574709110156744568720, −29.06202152282883534604957332993, −27.44851589227244838085762932813, −26.055986206870119875987573857923, −25.48843373930479483091799136060, −24.13619094897060529534994626872, −23.71470130200581180653016699055, −22.65231285955452618147643069948, −20.9482322509620477373591462480, −20.45279161677368304454894012806, −18.55329688786742361136181137012, −17.80829137081405060072653410662, −16.57734438300851248478385570443, −15.11911511688602089976419997843, −14.21613119259142685684588732465, −13.24069993449634742502087014922, −12.28803753435326842102581603641, −10.89897397970350352416248758088, −8.57939862484846384694597903508, −7.84135874347058019396009530823, −6.68181675964789571663247628410, −5.34061663836385866024686857837, −3.76419832603733302869413867023, −2.06187526983940062049471052676, 2.12546154721162117462421908975, 3.51960320186130940922663834947, 4.79368846975929186817080207377, 5.802436911804152658284215225854, 8.16296604441608108520879483008, 9.37896889762086753656117937127, 10.79998806867522949166893918633, 11.344006276474443448587050326540, 13.08356552318137296016829012894, 14.11239157732560538045480587893, 15.18249402699741692941898275785, 15.98060207927602600067977877577, 17.84570488618352013489091757834, 19.08236163749877058483271099862, 20.31121920311471592279419930276, 21.21982723124036983664956009698, 21.670406370401951927979227240323, 23.08959642678864234218189459513, 24.05260885325361216209344476222, 25.394828611829737738649726141525, 26.66339222723310376197979404009, 27.87762852487181916023008340622, 28.34311423678644085855771458282, 29.84790211333728756397624966002, 30.953967668553881168523691010751

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