L-function 1-119-119.6-r0-0-0

• Label: 1-119-119.6-r0-0-0
• Degree: $1$
• Conductor: $119$
• Sign: $-0.837 + 0.547i$
• Analytic cond.: $0.552633$
• Root an. cond.: $0.552633$
• 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 + (0.382 + 0.923i)5^-s + (−0.923 − 0.382i)6^-s + (−0.707 + 0.707i)8^-s + (0.707 − 0.707i)9^-s + (−0.382 + 0.923i)10^-s + (−0.923 − 0.382i)11^-s + (−0.382 − 0.923i)12^-s + i·13^-s + (−0.707 − 0.707i)15^-s − 16^-s + 18^-s + (−0.707 − 0.707i)19^-s + (−0.923 + 0.382i)20^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$119$    =    $7 \cdot 17$
Sign:	$-0.837 + 0.547i$
Analytic conductor:	$0.552633$
Root analytic conductor:	$0.552633$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{119} (6, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 119,\ (0:\ ),\ -0.837 + 0.547i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.3051174217 + 1.024455680i$
$L(1)$	$\approx$	$0.7696574886 + 0.7856176571i$

Euler product

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

	$p$	$F_p(T)$
bad	7	$1$
	17	$1$
good	2	$1 + (0.707 + 0.707i)T$
	3	$1 + (-0.923 + 0.382i)T$
	5	$1 + (0.382 + 0.923i)T$
	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

−28.79070472558517166282218748921, −28.17389707179300093345074642076, −27.213036997454679282010474818889, −25.17339117449492446344633289800, −24.47603604057743674868557671008, −23.31720473672543548342433352806, −22.837610483617573696785639800041, −21.44281856868729394940114383641, −20.821321678635067728863531651683, −19.60799540699056778020619013803, −18.4045168506845615608586417952, −17.459277861528155450055261375385, −16.22254030658435231997424240109, −15.08511929331746285229238364334, −13.378010136822308845406248022362, −12.855125191271579429324484962544, −11.99099827843419773070615786398, −10.672261055483333086470689326522, −9.86756509188240645822975513780, −8.0750664319442218718092730087, −6.30869555046806777906701195417, −5.32859378311603003478821801512, −4.484942650508736821385135526000, −2.435695877007640825270315520288, −0.94226626654568488407049412610, 2.710743645300153955966141937101, 4.21344285000112689542449543791, 5.42593378975691107593390805951, 6.43469033814034479184759804130, 7.31709464817159516905941144260, 9.103979924250952098831172687299, 10.64576823112022882195086097694, 11.4653915106826299222953840908, 12.80898724411866776705321553768, 13.89858471466087487461495842611, 15.066930211938604178167683393464, 15.910758678693970966186168870503, 17.00322562398669996567434020770, 17.86648716236631204309483531110, 18.924055770431924988909492549204, 21.0958913237535286996821140067, 21.54588568541872962272474609959, 22.510506411295029000170482782884, 23.42556525777692757799304970331, 24.108451744570359839141313886838, 25.61795311758925163699546443217, 26.39392496602285169255548171621, 27.221225329299839092866179294211, 28.791049900298096151263146058586, 29.541148792128097610381720385191

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