L-function 1-85-85.14-r1-0-0

• Label: 1-85-85.14-r1-0-0
• Degree: $1$
• Conductor: $85$
• Sign: $0.968 + 0.250i$
• Analytic cond.: $9.13451$
• Root an. cond.: $9.13451$
• 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 + (0.923 + 0.382i)3^-s − i·4^-s + (−0.923 + 0.382i)6^-s + (−0.382 − 0.923i)7^-s + (0.707 + 0.707i)8^-s + (0.707 + 0.707i)9^-s + (0.923 − 0.382i)11^-s + (0.382 − 0.923i)12^-s − i·13^-s + (0.923 + 0.382i)14^-s − 16^-s − 18^-s + (0.707 − 0.707i)19^-s − i·21^-s + (−0.382 + 0.923i)22^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$85$    =    $5 \cdot 17$
Sign:	$0.968 + 0.250i$
Analytic conductor:	$9.13451$
Root analytic conductor:	$9.13451$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{85} (14, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 85,\ (1:\ ),\ 0.968 + 0.250i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.620496926 + 0.2060868501i$
$L(1)$	$\approx$	$1.091463073 + 0.2189730300i$

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	$1 + (0.923 + 0.382i)T$
	7	$1 + (-0.382 - 0.923i)T$
	11	$1 + (0.923 - 0.382i)T$
	13	$1 - iT$
	19	$1 + (0.707 - 0.707i)T$
	23	$1 + (0.923 - 0.382i)T$
	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

−30.50137809946431807577679705447, −29.347230512710687804304692606335, −28.443636219015225317633302403750, −27.24067786911380157114926879544, −26.26608779827375332313823359348, −25.28131392605584958940622905024, −24.58927237800833062326956830963, −22.69763091512576131073520326916, −21.50023381467940726086165558412, −20.616163287440834185305705265366, −19.35665624053114556991738633945, −18.92017115024898258357114622317, −17.73535013545848440726856623564, −16.34838266138238432248137448924, −14.99354924254588020921706455392, −13.65021053507424701551909866705, −12.42243845807512292983898712950, −11.59176581612154845945933617608, −9.612709026336354630943416879009, −9.15693233760104651333120648813, −7.85224886221858005304273809957, −6.562445535207692616448961634566, −4.04727258395033041439880221760, −2.71743368480998168859905912093, −1.47483216753465449327665106128, 1.03870173904658438839804357666, 3.204096725744895456911218453034, 4.82308053022495394980421490358, 6.64828752562168549730907897057, 7.73614137240779641774672885018, 8.91401043749632349233098804726, 9.905999461304378219325073573910, 10.95711067511358172125007275140, 13.23348094126034354772284250729, 14.204020699161438058609645765165, 15.22318465233910523970906578917, 16.30338754271236377669722668201, 17.26084327709664899062428242478, 18.68049001396662027587529106491, 19.83782811183154708933930346881, 20.27621877178763385621998108624, 22.04633780290280853037513809752, 23.25310497638154336094244995141, 24.612506175870748912618936410381, 25.27905223817930811874722512559, 26.46103776122229073514320960767, 26.98949575585431324936661935824, 28.02097913405855246697954647338, 29.484219225704698038196085904888, 30.50391780400508807247516042834

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