Properties

Label 1-11-11.3-r0-0-0
Degree $1$
Conductor $11$
Sign $0.794 - 0.606i$
Analytic cond. $0.0510837$
Root an. cond. $0.0510837$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(11\)
Sign: $0.794 - 0.606i$
Analytic conductor: \(0.0510837\)
Root analytic conductor: \(0.0510837\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 11,\ (0:\ ),\ 0.794 - 0.606i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4586229999 - 0.1550664788i\)
\(L(\frac12)\) \(\approx\) \(0.4586229999 - 0.1550664788i\)
\(L(1)\) \(\approx\) \(0.7250693190 - 0.1998680382i\)
\(L(1)\) \(\approx\) \(0.7250693190 - 0.1998680382i\)

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

−45.39463594395501975969744807702, −44.56765680602640214369468348421, −42.899090013539892090284322164020, −41.23125931567637554417735242172, −40.60933246340791064114988799240, −39.08913821171983707731225893848, −36.30546533630583691375233672495, −35.46982735482229339989370478080, −34.09991038060261382498068251526, −32.66700911823403514660581948907, −31.25865240735452785918819107938, −29.20335415146647226619136605636, −27.84377795446691855174647964583, −25.56417383676554059016327182044, −24.39768465362899944342326661126, −23.10346608245067381762829290284, −21.582011399172421619050999930936, −18.75843000584654520807742707215, −17.08321428147369679317872020542, −16.00036570903338122322075620004, −13.458305030994073524444602358088, −12.24620851989253075343167730192, −9.005712909958607941868114994670, −6.70621979183662888694755223132, −5.13369962695377616120259659427, 3.610040431481681825088887609117, 6.031809302694153993116837471997, 9.968986597464014358784736223774, 10.91936691398131290146712763749, 12.936436403843608731167774240940, 14.99397619489517942605272840683, 17.33108585992351038226176172724, 19.009840214653417017485996057466, 20.971032806510044860068813509656, 22.421707347750038595969581921863, 23.19330453707956327421809503410, 26.248122472265924935457925061771, 27.67656102096692136867646567071, 29.21369696511081988117988557045, 30.14740142621094175361038152744, 32.27324327208079803820297211333, 33.40414126952326206289051904017, 35.23358256653947003802658538966, 37.2745777994939241434217316583, 38.537390547170970142134678797481, 39.47811011858490619376853346972, 40.928978128840741705820733145210, 42.2813381913511558130191347551, 44.87521159907991242916728260552, 45.63392199506539227555534305144

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