L-function 1-1140-1140.227-r0-0-0

• Label: 1-1140-1140.227-r0-0-0
• Degree: $1$
• Conductor: $1140$
• Sign: $0.525 - 0.850i$
• Analytic cond.: $5.29413$
• Root an. cond.: $5.29413$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

− i·7^-s + 11^-s + i·13^-s − i·17^-s − i·23^-s − 29^-s + 31^-s − i·37^-s + 41^-s + i·43^-s + i·47^-s − 49^-s − i·53^-s + 59^-s + 61^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1140$    =    $2^{2} \cdot 3 \cdot 5 \cdot 19$
Sign:	$0.525 - 0.850i$
Analytic conductor:	$5.29413$
Root analytic conductor:	$5.29413$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1140} (227, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1140,\ (0:\ ),\ 0.525 - 0.850i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.336266868 - 0.7450175743i$
$L(1)$	$\approx$	$1.108350007 - 0.2098103474i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	3	$1$
	5	$1$
	19	$1$
good	7	$1$
	11	$1$
	13	$1$
	17	$1$
	23	$1$
	29	$1$
	31	$1$
	37	$1 + T$
	41	$1$

Imaginary part of the first few zeros on the critical line

−21.575011138846668712224732143396, −20.6835119289548769772196409706, −19.79834918808183066046926059757, −19.17600527876318025805194026695, −18.405781214861395421848575686885, −17.46551858529879389703087303229, −17.02002837784977385221616335621, −15.831909437712958714914164881991, −15.19708966546148339305869957400, −14.64302852648811914873356827816, −13.55759226123949781055492987465, −12.76388708071169163515825909418, −11.99703338522357092385887001589, −11.33432967237147793201900532704, −10.29592639816103086454174044716, −9.4774010736928884397553149605, −8.65760608694122888000912995183, −7.96407687614091383131749858434, −6.86171980445668254924658914292, −5.91593191757778259348781863408, −5.370132129948515283360612070291, −4.11251928264741153911859287473, −3.27081130072115556938431777687, −2.21485891047865645352248187746, −1.184972468918150498289309812, 0.71165310093034184665319250383, 1.783045237148080291511724263818, 2.97451212993058949247932467398, 4.14265329594566012291459555743, 4.51554491526329991881098964057, 5.882587718525049503828577341616, 6.81952349427560957588989580341, 7.30821275530483636605130780183, 8.44648330654165149349784435159, 9.35392559139330061841927476510, 9.9595352214427649598442903143, 11.11291555444774113617915262578, 11.556756101634639531322049435265, 12.58949752490716011971166919175, 13.46438966619363924779848887805, 14.31622727037992969868774523576, 14.61013330133643177659282021498, 16.11314613817821335740406144884, 16.424340368834563620282092955261, 17.312182698563445647183059369229, 17.991750515137432745135968883506, 19.11393833476442836157969796194, 19.51855262684559421183720665281, 20.58553473059226600797214372005, 20.933810409652128191331577434664

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