L-function 1-119-119.109-r1-0-0

• Label: 1-119-119.109-r1-0-0
• Degree: $1$
• Conductor: $119$
• Sign: $0.100 - 0.994i$
• 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

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

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$119$    =    $7 \cdot 17$
Sign:	$0.100 - 0.994i$
Analytic conductor:	$12.7883$
Root analytic conductor:	$12.7883$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{119} (109, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 119,\ (1:\ ),\ 0.100 - 0.994i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.720098173 - 1.554331372i$
$L(1)$	$\approx$	$1.472723223 - 0.4767351903i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−29.16801669056498336048110030468, −28.74100205799734592693196778714, −26.82311942914898094086080311872, −25.8255183088102727668875981294, −24.88983979338560899174785463331, −23.6677531217015613348495772440, −23.27716899242652039122050965864, −22.08670686925369609563848906490, −21.43716015894558612311397081563, −19.872275793941314999800879647946, −18.67196365114659127681150016036, −17.792771839345826223669749696149, −16.50098863609834257656982838410, −15.508921571637078432105064908845, −14.05736632565921326283491823856, −13.569351069199547294765031071461, −12.17172442922332550066204704394, −11.37394296190668425562450452014, −10.29702087520379520656293763639, −7.95199792305535054846853346973, −7.02145226324608095830103707247, −6.10951113217221445928760850807, −4.98179373466007111977717214801, −3.19360326635810246796171950153, −1.931439818847609255593539439530, 0.746558556289842788367235680184, 2.835170432845601337968433312570, 4.36957418383459697669089614833, 5.1891613699808632607166857857, 6.09242850834953851671974829246, 7.9624232658125126299746127164, 9.699389679827234078275723213117, 10.5546339809542895853550317341, 11.82541420987066567382354352241, 12.68177643484710917579464909065, 13.749530372918222962177706333539, 15.293435271708766231903762225472, 15.88030167644824535526966746333, 16.90493327538904391962999617585, 18.187502992823092690430523949528, 20.07218763957322466833231035318, 20.593170379761345576461541554799, 21.51493053765241007650049030098, 22.49189013317486573003052569742, 23.410528388434516398352204572357, 24.289953302921034107166734788748, 25.377166115023802524839911866700, 26.75704266686276185609780502718, 28.11203849855328271827319270797, 28.58018145914246863004607892997

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