L-function 1-119-119.89-r1-0-0

• Label: 1-119-119.89-r1-0-0
• Degree: $1$
• Conductor: $119$
• Sign: $0.0125 - 0.999i$
• 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

+ (0.5 + 0.866i)2^-s + (−0.866 − 0.5i)3^-s + (−0.5 + 0.866i)4^-s + (0.866 − 0.5i)5^-s − i·6^-s − 8^-s + (0.5 + 0.866i)9^-s + (0.866 + 0.5i)10^-s + (−0.866 − 0.5i)11^-s + (0.866 − 0.5i)12^-s − 13^-s − 15^-s + (−0.5 − 0.866i)16^-s + (−0.5 + 0.866i)18^-s + (−0.5 − 0.866i)19^-s + i·20^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0125 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$119$    =    $7 \cdot 17$
Sign:	$0.0125 - 0.999i$
Analytic conductor:	$12.7883$
Root analytic conductor:	$12.7883$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{119} (89, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 119,\ (1:\ ),\ 0.0125 - 0.999i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.5390977936 - 0.5323690786i$
$L(1)$	$\approx$	$0.8629306776 + 0.08801127480i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−29.32852352994478508488495955768, −28.395432685078672259708055910515, −27.3466360502849648224745261091, −26.41850552568678350619327914016, −24.95265826591282051386210702491, −23.546785475132513249653682015656, −22.8745041387734729371258481394, −21.78126201335404513713907990136, −21.33688898770155241765228061103, −20.20846423922868559994396445160, −18.69405189936084554088320464535, −17.92207008057654872851682699803, −16.877090952331693089361576451165, −15.27631632382868274758112749827, −14.45702832425211500549958789944, −13.04113253657651144860695166025, −12.20362320494196037454109524165, −10.830636977799984371001973453327, −10.2353344664844209892457594669, −9.29153904493803615677201539375, −6.94069414738572563205149576068, −5.596813378671643683519230813388, −4.85966682528575309462425404081, −3.23967777296623232279017196724, −1.767110132073562582086178043191, 0.28057690808344899328597376859, 2.45602213262898216197384625918, 4.71939496929002543361502803290, 5.45342054576510226874754801142, 6.51740646616702729867921458241, 7.64634473020185737666162001540, 8.99830498529230219534167364274, 10.49191355935487874278645651354, 12.01628456717477105330682762567, 13.032585872649971187933388903766, 13.613482898952261339168134476818, 15.09545750845779603388107673837, 16.377038719930329215943301038561, 17.11622709987852279658888128734, 17.85921202395113371960220841070, 19.00764587816758592605668593860, 20.88828175687849239172021163009, 21.7781430930872468270364280028, 22.59134321909181379376832423759, 23.85713350127712696468165499049, 24.33793587183253129717557589839, 25.293842913613816213676836341371, 26.39526874548868472598853288485, 27.60539008314541064986727693042, 28.84874511444154468782630494190

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