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Properties

Label 1-11-11.8-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 $0$

Origins

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Normalization:  

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 + ⋯
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

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

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: $\chi_{11} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 11,\ (1:\ ),\ -0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3214164081 + 0.6890281810i\)
\(L(\frac12)\) \(\approx\) \(0.3214164081 + 0.6890281810i\)
\(L(1)\) \(\approx\) \(0.5422944263 + 0.5305008734i\)
\(L(1)\) \(\approx\) \(0.5422944263 + 0.5305008734i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T \)
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 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + T \)
97 \( 1 + (0.309 - 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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