L-function 1-11-11.5-r0-0-0

• Label: 1-11-11.5-r0-0-0
• Degree: $1$
• Conductor: $11$
• Sign: $0.624 + 0.781i$
• Analytic cond.: $0.0510837$
• Root an. cond.: $0.0510837$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.809 + 0.587i)2^-s + (0.309 + 0.951i)3^-s + (0.309 − 0.951i)4^-s + (−0.809 − 0.587i)5^-s + (−0.809 − 0.587i)6^-s + (0.309 − 0.951i)7^-s + (0.309 + 0.951i)8^-s + (−0.809 + 0.587i)9^-s + 10^-s + 12^-s + (−0.809 + 0.587i)13^-s + (0.309 + 0.951i)14^-s + (0.309 − 0.951i)15^-s + (−0.809 − 0.587i)16^-s + (−0.809 − 0.587i)17^-s + (0.309 − 0.951i)18^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$11$
Sign:	$0.624 + 0.781i$
Analytic conductor:	$0.0510837$
Root analytic conductor:	$0.0510837$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{11} (5, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 11,\ (0:\ ),\ 0.624 + 0.781i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.3561278323 + 0.1713091896i$
$L(1)$	$\approx$	$0.5805311136 + 0.2129383511i$

Euler product

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

	$p$	$F_p(T)$
bad	11	$1$
good	2	$1 + (-0.809 + 0.587i)T$
	3	$1 + (0.309 + 0.951i)T$
	5	$1 + (-0.809 - 0.587i)T$
	7	$1 + (0.309 - 0.951i)T$
	13	$1 + (-0.809 + 0.587i)T$
	17	$1 + (-0.809 - 0.587i)T$
	19	$1 + (0.309 + 0.951i)T$
	23	$1 + T$
	29	$1 + (0.309 - 0.951i)T$

Imaginary part of the first few zeros on the critical line

−45.992857876740975256429899631195, −44.166582569894431073202390852504, −42.73776464425257506936753077255, −41.21327507583455505413149754563, −39.44319270094730994698826415607, −37.8879189468480404229255113447, −36.830721606501950842779862714074, −35.103589002333235032242136313328, −34.5972204305548442140229585985, −31.32845534136625909536474896094, −30.50526927558957657974996791575, −28.9996374398845296933700159472, −27.39714864997876724666794402141, −25.85623737581020255318374607184, −24.33661634366136980503787359604, −22.15714139133311048098743192458, −19.9906260243439205467866780351, −18.89344173569352104267570432504, −17.68807603791082994478424454831, −15.169105282641905401132167592826, −12.62329617142335150431012578865, −11.27579624197347705420756999010, −8.704161065911558045022104093621, −7.20692647129910428845946072570, −2.69600408486917233690482283449, 4.62935366251112260134094795549, 7.66185762868686205184568530025, 9.32576278595562580840034682186, 11.08604090734966310677523591806, 14.391345209913737768557754830718, 15.95701473940678460989492756920, 17.0589877129008589583878475262, 19.50752880683188820541576949914, 20.61265054372333079178400924558, 23.13907860578284012046137252354, 24.72083578855277620657306332268, 26.68452916273036899287058730843, 27.22591812668902221184551478255, 28.83880063287434417342469151843, 31.42427523311059427895530059687, 32.83760075474434132336180921300, 33.99262875889834913169724447328, 35.76575289158548786947394304876, 36.92818035598601603325885144168, 38.48574617295503178661726067050, 39.76404580393091217961458428396, 42.30708641995038782073842321800, 43.36021389069667485514209185454, 44.1807543932131508618010042867, 45.75877859475394468803196671809

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