L-function 1-119-119.117-r1-0-0

• Label: 1-119-119.117-r1-0-0
• Degree: $1$
• Conductor: $119$
• Sign: $0.998 + 0.0590i$
• 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

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

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0590i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$119$    =    $7 \cdot 17$
Sign:	$0.998 + 0.0590i$
Analytic conductor:	$12.7883$
Root analytic conductor:	$12.7883$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{119} (117, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 119,\ (1:\ ),\ 0.998 + 0.0590i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$3.802318553 + 0.1123720924i$
$L(1)$	$\approx$	$2.249922015 + 0.01623216431i$

Euler product

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

	$p$	$F_p(T)$
bad	7	$1$
	17	$1$
good	2	$1 + (0.866 - 0.5i)T$
	3	$1 + (0.258 + 0.965i)T$
	5	$1 + (0.965 + 0.258i)T$
	11	$1 + (0.965 - 0.258i)T$
	13	$1 + T$
	19	$1 + (-0.866 + 0.5i)T$
	23	$1 + (-0.258 + 0.965i)T$
	29	$1 + (0.707 - 0.707i)T$
	31	$1 + (0.258 + 0.965i)T$

Imaginary part of the first few zeros on the critical line

−29.2366490624898800479386476028, −28.13353689705530485483042961213, −26.25680758779544932958310020641, −25.45133946390120605838160475134, −24.854325136060993826988515909827, −23.91546815231340122513309875537, −22.935769522761471634705533235577, −21.86555812538144399943570225192, −20.77013703202387080386049288933, −19.86320303243607646138724839638, −18.31291123151834877830404535156, −17.38324301360664024745910190366, −16.497484763916757985216837930295, −14.87693767438535531381182450784, −14.04409980411712565298603421256, −13.172295221397501629675902723260, −12.38903253319277973226570177910, −11.10428734508036942819373408730, −9.13018481223440245253939651486, −8.11691171499340237101353818252, −6.52931804179828686321094884738, −6.16119781856090935375483819544, −4.48609168875075066949586434812, −2.82173523928539309258399240640, −1.50730351107540076491588590122, 1.66964190584675517191055878309, 3.148820570718655601170781698319, 4.19207712077848890037897358089, 5.58587414013420882282616172723, 6.464128784606674833106805852351, 8.74093621045659349462629139347, 9.86813501674525895398427633486, 10.70794646603880999468173880237, 11.78510280070205889850113020509, 13.403653109877277378084340669527, 14.10613346725756424340116207224, 15.02718521769007960706347601356, 16.15346103657557562082884745544, 17.34749077030737839873596098040, 18.8991628080248716450566505410, 19.99341465326220808863649552053, 21.05383118398341826301861977989, 21.61549298514984109089162130269, 22.47632427498947247247283733089, 23.48540169722834527554402419556, 25.095228016518249830640386935788, 25.544786448591587776329008713694, 27.042582252319064947215122728131, 28.032471254572960007633677297586, 28.94957905348786567382095993260

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