L-function 1-152-152.107-r0-0-0

• Label: 1-152-152.107-r0-0-0
• Degree: $1$
• Conductor: $152$
• Sign: $-0.0977 - 0.995i$
• Analytic cond.: $0.705885$
• Root an. cond.: $0.705885$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.5 − 0.866i)3^-s + (0.5 − 0.866i)5^-s − 7^-s + (−0.5 − 0.866i)9^-s + 11^-s + (−0.5 − 0.866i)13^-s + (−0.5 − 0.866i)15^-s + (−0.5 + 0.866i)17^-s + (−0.5 + 0.866i)21^-s + (0.5 + 0.866i)23^-s + (−0.5 − 0.866i)25^-s − 27^-s + (−0.5 − 0.866i)29^-s + 31^-s + (0.5 − 0.866i)33^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$152$    =    $2^{3} \cdot 19$
Sign:	$-0.0977 - 0.995i$
Analytic conductor:	$0.705885$
Root analytic conductor:	$0.705885$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{152} (107, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 152,\ (0:\ ),\ -0.0977 - 0.995i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.8257953144 - 0.9108661640i$
$L(1)$	$\approx$	$1.037890799 - 0.5531722440i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	19	$1$
good	3	$1 + (0.5 - 0.866i)T$
	5	$1 + (0.5 - 0.866i)T$
	7	$1 - T$
	11	$1 + T$
	13	$1 + (-0.5 - 0.866i)T$
	17	$1 + (-0.5 + 0.866i)T$
	23	$1 + (0.5 + 0.866i)T$
	29	$1 + (-0.5 - 0.866i)T$
	31	$1 + T$

Imaginary part of the first few zeros on the critical line

−28.31434410357549105647708965280, −26.890206944033573488142121358115, −26.544786471549733585429670152830, −25.464180592597059360483414001276, −24.79634870152335305262641967242, −23.00023276838663618871987257960, −22.19478800047481299443907176951, −21.68772346450555481787902989593, −20.35762191198971393820385416683, −19.43698558965462839403091608217, −18.57001026705757354528684544829, −17.05568877817173706166414399073, −16.285661095789286280355040669866, −15.06946106851121444413570225761, −14.27635989363582612330175321585, −13.3926776014099634401444654854, −11.772321501525744549986596394177, −10.61632349284876406755844270679, −9.58682520765520450910480777443, −9.01542667845017877890508971854, −7.14812835749453610040334029642, −6.24056874369908183306854486676, −4.61513805801410232939481757996, −3.36484014326040880869171846597, −2.357642066886381512117172695111, 1.06790722963546637743626535438, 2.51881649374690935933780053936, 3.936167368928725345604960953830, 5.736126303687456700849377301554, 6.62900419327903790987741168732, 7.97784362168114342715596410111, 9.08263412975814841636890051849, 9.86035875503758766117611610379, 11.7162486212093102694749646002, 12.84031014063003643301509961058, 13.21784727048647127852669593264, 14.49060601031081352964076363451, 15.667993637981657776875857236706, 17.10054543238903522316281414843, 17.58071935703127549766428130732, 19.17441800183674730967353386885, 19.69059158638698073741022898677, 20.615289045881211850662105630252, 21.916582061506736551188636802431, 22.94868727961195349525175931236, 24.12256154247257722431751255026, 24.96368428950826514145678730591, 25.48430618554782836023482770258, 26.61520505364004980188404024730, 27.97609404333288980005199467299

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