L-function 1-11-11.7-r1-0-0

• Label: 1-11-11.7-r1-0-0
• Degree: $1$
• Conductor: $11$
• Sign: $-0.642 - 0.766i$
• Analytic cond.: $1.18211$
• Root an. cond.: $1.18211$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

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

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$11$
Sign:	$-0.642 - 0.766i$
Analytic conductor:	$1.18211$
Root analytic conductor:	$1.18211$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{11} (7, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 11,\ (1:\ ),\ -0.642 - 0.766i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.3214164081 - 0.6890281810i$
$L(1)$	$\approx$	$0.5422944263 - 0.5305008734i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−45.61383532443762936966007233188, −44.50315105809981977478377956073, −43.28237879087022285362370498164, −41.451985024130012558407220182013, −40.50913716343137847444457958349, −38.40599013182803415397550119015, −37.07569800336932558076454614719, −35.01510928430873838470872938337, −33.99140302987517963033014866190, −33.178842589001558746303926414737, −31.22940901204067250687963009483, −28.947771705705823981195394585, −27.40725717753448042762789920871, −26.328449037768254749252262762333, −24.45819581215308996084408240869, −22.83885406338058962719853601162, −21.59808825881419508444894553380, −18.527889377871107268152218354111, −17.43161793522573674272782938011, −15.68125120531344955254954835691, −14.31664759932336707218166025316, −11.259986049139631879002309758061, −9.42882532201290964105489786007, −6.85188812449087884224022796653, −5.07031637930817115271739478905, 1.23118824094644557330126632913, 4.9627151638056233330962205550, 8.08939114538406115212311784238, 10.4500536382022332221178893649, 12.11506980690332170753638554345, 13.437657526437096729270089409681, 16.94978978223110727613037943840, 17.87481329763534279900237596385, 19.83012856053316642706724002179, 21.32039371631114630973784591369, 23.103365240819911054545433311199, 24.71060202237696255086224193381, 27.19549761157675669135899555136, 28.41349834370446234328316226414, 29.57219606296392511801120043627, 30.86131192335796380131399489359, 32.88578028689798316267070256069, 34.89957117724815985971476450101, 36.20301087650878318423744955110, 37.23130886007165567642745472163, 39.57532585824936030442348526707, 39.955595985887364149097406429137, 41.40118442396643591924427168519, 43.84491852037660538321332748361, 45.18664587132294167255369395899

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