L-function 1-119-119.39-r1-0-0

• Label: 1-119-119.39-r1-0-0
• Degree: $1$
• Conductor: $119$
• Sign: $-0.225 - 0.974i$
• Analytic cond.: $12.7883$
• Root an. cond.: $12.7883$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.258 − 0.965i)2^-s + (0.991 − 0.130i)3^-s + (−0.866 + 0.5i)4^-s + (0.793 − 0.608i)5^-s + (−0.382 − 0.923i)6^-s + (0.707 + 0.707i)8^-s + (0.965 − 0.258i)9^-s + (−0.793 − 0.608i)10^-s + (0.608 − 0.793i)11^-s + (−0.793 + 0.608i)12^-s + i·13^-s + (0.707 − 0.707i)15^-s + (0.5 − 0.866i)16^-s + (−0.5 − 0.866i)18^-s + (−0.258 − 0.965i)19^-s + (−0.382 + 0.923i)20^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$119$    =    $7 \cdot 17$
Sign:	$-0.225 - 0.974i$
Analytic conductor:	$12.7883$
Root analytic conductor:	$12.7883$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{119} (39, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 119,\ (1:\ ),\ -0.225 - 0.974i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.483265398 - 1.866674439i$
$L(1)$	$\approx$	$1.214081013 - 0.8432104931i$

Euler product

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

	$p$	$F_p(T)$
bad	7	$1$
	17	$1$
good	2	$1 + (-0.258 - 0.965i)T$
	3	$1 + (0.991 - 0.130i)T$
	5	$1 + (0.793 - 0.608i)T$
	11	$1 + (0.608 - 0.793i)T$
	13	$1 + iT$
	19	$1 + (-0.258 - 0.965i)T$
	23	$1 + (0.991 + 0.130i)T$
	29	$1 + (0.923 + 0.382i)T$
	31	$1 + (-0.991 + 0.130i)T$

Imaginary part of the first few zeros on the critical line

−29.1950833370495878655847298102, −27.64234371831753463098288182031, −26.9864628211382598374521401634, −25.82448990859460349391757126692, −25.29112159792188229091906770095, −24.642757450176559362523851951761, −23.065799783086659304326800542433, −22.22297606680319633807617068053, −20.99409704119822077329845960096, −19.76630421695744437166960192564, −18.67860666658343085521763780588, −17.80253777921432857156093451735, −16.730702683980873751941272631464, −15.23466370955151568002831266773, −14.74518822294305557413392122876, −13.755408948542888177438073589436, −12.76149081058950250854290388201, −10.40186149272452179532429637107, −9.71791843744150836523579239402, −8.58831306529615990806890090699, −7.4209877904138599971861561009, −6.40128353484051748759701272665, −4.92910128314827657746475862006, −3.370042957051767265435378476797, −1.65407866589513028391805883117, 1.12447605245480824909521371182, 2.25906778262695269920852603274, 3.608346824971820732572987709974, 4.92746564605804716232209571155, 6.894771127347114405516804066965, 8.742411789880329852636793625132, 8.95496611814468944694838839721, 10.18694081952406863687482066396, 11.597889702305038053038166922180, 12.89077308942057555695063259222, 13.644547792616944758860867965630, 14.48985283244564122190046604096, 16.32546008524707026971668701337, 17.389378540213539948158755073947, 18.58278741021318373363145958215, 19.48426238997443170875735544477, 20.300391051731453516441468252865, 21.48967069750885966335875953637, 21.70413490816358280429986564197, 23.61042693763526256419313862383, 24.717278019410623169650403482, 25.7266444787959404613286806753, 26.623065733998024163430498749750, 27.592733136664840974737607910917, 28.80986382805727912076158297835

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