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Properties

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 $0$

Origins

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Normalization:  

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 + ⋯
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

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

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

\(L(\frac{1}{2})\) \(\approx\) \(0.3561278323 + 0.1713091896i\)
\(L(\frac12)\) \(\approx\) \(0.3561278323 + 0.1713091896i\)
\(L(1)\) \(\approx\) \(0.5805311136 + 0.2129383511i\)
\(L(1)\) \(\approx\) \(0.5805311136 + 0.2129383511i\)

Euler product

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

−45.992857876740975256429899631195, −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