LMFDB - L-function 1-197-197.44-r1-0-0

Dirichlet series

L(s) = 1

+ (0.545 + 0.838i)2^-s + (−0.695 − 0.718i)3^-s + (−0.404 + 0.914i)4^-s + (0.886 + 0.462i)5^-s + (0.222 − 0.974i)6^-s + (0.838 + 0.545i)7^-s + (−0.987 + 0.159i)8^-s + (−0.0320 + 0.999i)9^-s + (0.0960 + 0.995i)10^-s + (−0.855 − 0.518i)11^-s + (0.938 − 0.345i)12^-s + (0.958 − 0.284i)13^-s + i·14^-s + (−0.284 − 0.958i)15^-s + (−0.672 − 0.740i)16^-s + (0.598 + 0.801i)17^-s + ⋯

L(s) = 1

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.647 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}

\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.647 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree:	$1$
Conductor:	$197$
Sign:	$-0.647 + 0.761i$
Analytic conductor:	$21.1705$
Root analytic conductor:	$21.1705$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{197} (44, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 197,\ (1:\ ),\ -0.647 + 0.761i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.8522959617 + 1.842848933i$
$L(\frac12)$	$\approx$	$0.8522959617 + 1.842848933i$
$L(1)$	$\approx$	$1.092085713 + 0.7148029936i$
$L(1)$	$\approx$	$1.092085713 + 0.7148029936i$

Euler product

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

	$p$	$F_p(T)$
bad	197	$1$
good	2	$1 + (0.545 + 0.838i)T$
	3	$1 + (-0.695 - 0.718i)T$
	5	$1 + (0.886 + 0.462i)T$
	7	$1 + (0.838 + 0.545i)T$
	11	$1 + (-0.855 - 0.518i)T$
	13	$1 + (0.958 - 0.284i)T$
	17	$1 + (0.598 + 0.801i)T$
	19	$1 + (-0.623 + 0.781i)T$
	23	$1 + (0.991 - 0.127i)T$
	29	$1 + (-0.345 - 0.938i)T$
	31	$1 + (-0.315 + 0.949i)T$
	37	$1 + (-0.672 + 0.740i)T$
	41	$1 + (-0.801 + 0.598i)T$
	43	$1 + (-0.518 + 0.855i)T$
	47	$1 + (-0.926 + 0.375i)T$
	53	$1 + (-0.761 + 0.648i)T$
	59	$1 + (-0.0960 + 0.995i)T$
	61	$1 + (0.718 + 0.695i)T$
	67	$1 + (-0.375 - 0.926i)T$
	71	$1 + (0.999 + 0.0320i)T$
	73	$1 + (0.740 + 0.672i)T$
	79	$1 + (0.886 - 0.462i)T$
	83	$1 + (-0.623 - 0.781i)T$
	89	$1 + (-0.315 - 0.949i)T$
	97	$1 + (0.981 - 0.191i)T$
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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

−26.63802314400280539495249274097, −25.47743668042770565047979427465, −23.955288767896818277295635463974, −23.48615731899361339231792059524, −22.459545184690609423726944248908, −21.35870020907437874172593524797, −20.86608138138665732942745597680, −20.34313425324157559690693988909, −18.53695664511905207093384126532, −17.79695181348891744976392593707, −16.85076108982681162624318009742, −15.605741354499748280247326509985, −14.537264665514480711053785397459, −13.50518513920021197606715855363, −12.62602591643476709778893293007, −11.32817380215234585092046489331, −10.69667957264026225185992443187, −9.752380563777443945153302903922, −8.7666519870188693617687556924, −6.728176839830113457764415612993, −5.24113991243822010762137663709, −4.980266804112118492186902827124, −3.620328161790926132393986875168, −1.952121467436539505333693471781, −0.6528912105179742010612249745, 1.60183096489217865887855166404, 3.06816446289142830573457026366, 4.98100652534855924683101306120, 5.78993503348002898466725195261, 6.43782477462389368980928635119, 7.83600494049011291503589010056, 8.55962992974394254739603107069, 10.43800043314904579276274862282, 11.40949998651755132584560311486, 12.68875650882599174610360280089, 13.37534670561919365834203723680, 14.34848289404098987478928110717, 15.33175529953188460548905686652, 16.62056388228581740739883686967, 17.38912514934850538908642422758, 18.31161724615146809616778265956, 18.759814678876493217762671990527, 21.14181684187409126592447041033, 21.33257024115895899971451108667, 22.607174867809703446445028471424, 23.3652183972799879641127655605, 24.19475494002121210801190757729, 25.12827171470066723446655772960, 25.66933930771671585431477640009, 26.9227165422100947868658818631

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Dirichlet series

Functional equation

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Euler product

Imaginary part of the first few zeros on the critical line

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

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1-197-197.44-r1-0-0
Degree	$1$
Conductor	$197$
Sign	$-0.647 + 0.761i$
Analytic cond.	$21.1705$
Root an. cond.	$21.1705$
Motivic weight	$0$
Arithmetic	yes
Rational	no
Primitive	yes
Self-dual	no
Analytic rank	$0$

Properties

Origins

Related objects

Downloads

Learn more

Particular Values

Imaginary part of the first few zeros on the critical line

Graph of the ZZZ-function along the critical line

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

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