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Properties

Label 1-11e2-121.5-r0-0-0
Degree $1$
Conductor $121$
Sign $0.443 - 0.896i$
Analytic cond. $0.561921$
Root an. cond. $0.561921$
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  + (−0.466 − 0.884i)2-s + (0.309 + 0.951i)3-s + (−0.564 + 0.825i)4-s + (0.198 − 0.980i)5-s + (0.696 − 0.717i)6-s + (−0.254 − 0.967i)7-s + (0.993 + 0.113i)8-s + (−0.809 + 0.587i)9-s + (−0.959 + 0.281i)10-s + (−0.959 − 0.281i)12-s + (0.941 − 0.336i)13-s + (−0.736 + 0.676i)14-s + (0.993 − 0.113i)15-s + (−0.362 − 0.931i)16-s + (0.974 − 0.226i)17-s + (0.897 + 0.441i)18-s + ⋯
L(s)  = 1  + (−0.466 − 0.884i)2-s + (0.309 + 0.951i)3-s + (−0.564 + 0.825i)4-s + (0.198 − 0.980i)5-s + (0.696 − 0.717i)6-s + (−0.254 − 0.967i)7-s + (0.993 + 0.113i)8-s + (−0.809 + 0.587i)9-s + (−0.959 + 0.281i)10-s + (−0.959 − 0.281i)12-s + (0.941 − 0.336i)13-s + (−0.736 + 0.676i)14-s + (0.993 − 0.113i)15-s + (−0.362 − 0.931i)16-s + (0.974 − 0.226i)17-s + (0.897 + 0.441i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.443 - 0.896i$
Analytic conductor: \(0.561921\)
Root analytic conductor: \(0.561921\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ 0.443 - 0.896i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7577294507 - 0.4704479202i\)
\(L(\frac12)\) \(\approx\) \(0.7577294507 - 0.4704479202i\)
\(L(1)\) \(\approx\) \(0.8493326381 - 0.3108326092i\)
\(L(1)\) \(\approx\) \(0.8493326381 - 0.3108326092i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (-0.466 - 0.884i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (0.198 - 0.980i)T \)
7 \( 1 + (-0.254 - 0.967i)T \)
13 \( 1 + (0.941 - 0.336i)T \)
17 \( 1 + (0.974 - 0.226i)T \)
19 \( 1 + (0.516 - 0.856i)T \)
23 \( 1 + (0.841 + 0.540i)T \)
29 \( 1 + (-0.921 + 0.389i)T \)
31 \( 1 + (0.610 - 0.791i)T \)
37 \( 1 + (-0.0285 + 0.999i)T \)
41 \( 1 + (-0.985 + 0.170i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 + (0.897 - 0.441i)T \)
53 \( 1 + (-0.362 + 0.931i)T \)
59 \( 1 + (-0.985 - 0.170i)T \)
61 \( 1 + (-0.466 + 0.884i)T \)
67 \( 1 + (-0.142 + 0.989i)T \)
71 \( 1 + (0.0855 + 0.996i)T \)
73 \( 1 + (0.774 - 0.633i)T \)
79 \( 1 + (-0.870 - 0.491i)T \)
83 \( 1 + (-0.998 - 0.0570i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (0.198 + 0.980i)T \)
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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

−29.070152084538748855022429510175, −28.2709600659811320656546278440, −26.905383252000142892333328656551, −25.87263344557656755282781944809, −25.33577554925770485513520944372, −24.48714820445026977476465912974, −23.20277096782614932937553431704, −22.6459961739929278339757620471, −21.081639637533685487353695839546, −19.39401912296495690996765543470, −18.600165053468090743661019446294, −18.28050441008996170810547235594, −16.95684450847809574511004366102, −15.60957072168716271968663359992, −14.58888755400292012719573327344, −13.87167864764310223602862904582, −12.55239575407589435423905787223, −11.118627534736559811878652782993, −9.66521389929783625863529407105, −8.5605295628303568953075183879, −7.50354884319652002572675117757, −6.38443569394015566747638139821, −5.695349645022055296801049641068, −3.27384615552839194966935694788, −1.68063467536998925113147109796, 1.11627882480066378199498190753, 3.12022721916954846813836438161, 4.13787210581151861313937167130, 5.30180145885157899429436036759, 7.64512231975974928785643994631, 8.803288143499064623658576281181, 9.64712578307638380952627493362, 10.58660059183845282068253535909, 11.67688190574399687883127459645, 13.202725101731840660297484490327, 13.79448583326079044555382782792, 15.62135145149180754411603213884, 16.72162232590613953688061919781, 17.24565292502265081117593911475, 18.846114044639003753583939014339, 20.15646261105504432220946894600, 20.47564872148733166264870843740, 21.36476275546432009516176373340, 22.51094477637757184958509197920, 23.56958664092560609638323792693, 25.32832465431842322511788697305, 26.036063068457152250710676231527, 27.11739766220039694743161939558, 27.87327847604403492549781231623, 28.65486989181371540696107892966

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