L-function 1-119-119.108-r0-0-0

• Label: 1-119-119.108-r0-0-0
• Degree: $1$
• Conductor: $119$
• Sign: $-0.0713 + 0.997i$
• 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

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

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0713 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$119$    =    $7 \cdot 17$
Sign:	$-0.0713 + 0.997i$
Analytic conductor:	$0.552633$
Root analytic conductor:	$0.552633$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{119} (108, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 119,\ (0:\ ),\ -0.0713 + 0.997i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.4951267042 + 0.5318105403i$
$L(1)$	$\approx$	$0.6899323970 + 0.3353795698i$

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	$1 + (0.793 + 0.608i)T$
	5	$1 + (-0.991 - 0.130i)T$
	11	$1 + (0.130 + 0.991i)T$
	13	$1 + iT$
	19	$1 + (0.965 - 0.258i)T$
	23	$1 + (-0.793 + 0.608i)T$
	29	$1 + (0.382 + 0.923i)T$
	31	$1 + (-0.793 - 0.608i)T$

Imaginary part of the first few zeros on the critical line

−28.89658063062128614713407834824, −27.6138020165458738509415645418, −26.80851306892662534928031225482, −26.114607743125997431710130393874, −24.78795225532500638238144238641, −24.29555005416576221107600253432, −22.86587559162864487481712391319, −21.34076165679117855575886405454, −20.12572467827732231243273409924, −19.66309803377356959809862516596, −18.63795450653537831689618348821, −17.913391187506325463058585077300, −16.340868081960626918313657352145, −15.48998714871814172047316977708, −14.26187784325509520200835035552, −12.75864139811432739012308087178, −11.83425392594324267793638341745, −10.68820418125544644749117932660, −9.26918282745927556110812793510, −8.12323779737028498931978706412, −7.640675658421935268549346632, −6.23544825498695042963582002340, −3.677284947670670502513923902447, −2.75589075801952795210892531798, −0.90243495687237910624343506433, 1.91452019348737441147343256374, 3.55486516003972318036188014848, 4.940413189203700163937931200242, 7.00936607740722499657524584211, 7.83708398404989188084341418203, 8.98472861352680745592243650322, 9.79804886449151418993721281542, 11.09495813183395162589193222312, 12.16950497032075393985845615304, 14.06881327946060495186744176725, 15.11113028148837338536524507245, 15.87187407780925947284206584814, 16.71821595850917588184220308483, 18.18412931313489168886672780098, 19.27814278641791848620753242049, 20.022782022322673284110841954643, 20.73067233093361642054046778022, 22.17216486127496386029915326988, 23.64068521645688280336253280531, 24.51274566987994896727356042437, 25.77096633674278074607857798641, 26.30120272907496794925220023961, 27.43822181244991809728255940733, 27.891057396868815811900346350340, 29.07076653000384001786053886003

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