L-function 1-740-740.47-r0-0-0

• Label: 1-740-740.47-r0-0-0
• Degree: $1$
• Conductor: $740$
• Sign: $0.339 + 0.940i$
• Analytic cond.: $3.43654$
• Root an. cond.: $3.43654$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.866 − 0.5i)3^-s + (0.866 + 0.5i)7^-s + (0.5 + 0.866i)9^-s − 11^-s + (0.866 + 0.5i)13^-s + (0.866 − 0.5i)17^-s + (−0.5 + 0.866i)19^-s + (−0.5 − 0.866i)21^-s − i·23^-s − i·27^-s − 29^-s − 31^-s + (0.866 + 0.5i)33^-s + (−0.5 − 0.866i)39^-s + (−0.5 + 0.866i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.339 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$740$    =    $2^{2} \cdot 5 \cdot 37$
Sign:	$0.339 + 0.940i$
Analytic conductor:	$3.43654$
Root analytic conductor:	$3.43654$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{740} (47, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 740,\ (0:\ ),\ 0.339 + 0.940i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.7459037642 + 0.5237031186i$
$L(1)$	$\approx$	$0.8273529721 + 0.08329726345i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
	37	$1$
good	3	$1 + (-0.866 - 0.5i)T$
	7	$1 + (0.866 + 0.5i)T$
	11	$1 - T$
	13	$1 + (0.866 + 0.5i)T$
	17	$1 + (0.866 - 0.5i)T$
	19	$1 + (-0.5 + 0.866i)T$
	23	$1 - iT$
	29	$1 - T$
	31	$1 - T$

Imaginary part of the first few zeros on the critical line

−22.335129885926657135892985738594, −21.472592056797063698711103829544, −20.76055274676405034819013671613, −20.31114175618217494201308754968, −18.773709647677372100436592399759, −18.23078818259869530642320543421, −17.34801697225262640759550767452, −16.76551919783967131085054453678, −15.801943290396228423129313606328, −15.13990393276006633204453193809, −14.234700207353798804826489335087, −13.08418119169007199054498689764, −12.449683975321565119259173318066, −11.19076853453209915674633761509, −10.80951282597592369972219732484, −10.12289285225908264574689505963, −8.8768168978959276681005380176, −7.95564600565462521041618403498, −7.02368050494904989109398799494, −5.86374332789876007704386671598, −5.19362795091731012014049308019, −4.29366168719521681247044381473, −3.3403884971607796779568299387, −1.80788519274572333093428664590, −0.51275103133610689132904736194, 1.32887564459168370729566208941, 2.09716479624500992401870282667, 3.56684459556852822755316784018, 4.88138221422823514649412224443, 5.51896405605712612376999172572, 6.295666053799759068640613069991, 7.598963557138463151677490585797, 7.99584448277571387443680112684, 9.24378488620271841838231911602, 10.38476074272971842272507440071, 11.1841997005588756939413669782, 11.76680498289168964699340741926, 12.69665866912509040265265998828, 13.45558779211371948108246107724, 14.40591287160115506695216656681, 15.39594725775975142976029116013, 16.29205731552923601334001233609, 16.93051748974515463633385692800, 18.03652319785546932469836620161, 18.43414379587772527813784366529, 19.02607570648451997837527907501, 20.39590294517377800411907608730, 21.22368453378107454012127962052, 21.68999250129296700638831675786, 22.88077840849102870813811286774

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