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Properties

Label 1-11e2-121.119-r0-0-0
Degree $1$
Conductor $121$
Sign $-0.987 - 0.160i$
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  + (−0.998 − 0.0570i)2-s + (0.309 − 0.951i)3-s + (0.993 + 0.113i)4-s + (−0.466 − 0.884i)5-s + (−0.362 + 0.931i)6-s + (−0.921 − 0.389i)7-s + (−0.985 − 0.170i)8-s + (−0.809 − 0.587i)9-s + (0.415 + 0.909i)10-s + (0.415 − 0.909i)12-s + (−0.870 + 0.491i)13-s + (0.897 + 0.441i)14-s + (−0.985 + 0.170i)15-s + (0.974 + 0.226i)16-s + (0.941 − 0.336i)17-s + (0.774 + 0.633i)18-s + ⋯
L(s)  = 1  + (−0.998 − 0.0570i)2-s + (0.309 − 0.951i)3-s + (0.993 + 0.113i)4-s + (−0.466 − 0.884i)5-s + (−0.362 + 0.931i)6-s + (−0.921 − 0.389i)7-s + (−0.985 − 0.170i)8-s + (−0.809 − 0.587i)9-s + (0.415 + 0.909i)10-s + (0.415 − 0.909i)12-s + (−0.870 + 0.491i)13-s + (0.897 + 0.441i)14-s + (−0.985 + 0.170i)15-s + (0.974 + 0.226i)16-s + (0.941 − 0.336i)17-s + (0.774 + 0.633i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $-0.987 - 0.160i$
Analytic conductor: \(0.561921\)
Root analytic conductor: \(0.561921\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ -0.987 - 0.160i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03338288640 - 0.4138647681i\)
\(L(\frac12)\) \(\approx\) \(0.03338288640 - 0.4138647681i\)
\(L(1)\) \(\approx\) \(0.4331806435 - 0.3331070966i\)
\(L(1)\) \(\approx\) \(0.4331806435 - 0.3331070966i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (-0.998 - 0.0570i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (-0.466 - 0.884i)T \)
7 \( 1 + (-0.921 - 0.389i)T \)
13 \( 1 + (-0.870 + 0.491i)T \)
17 \( 1 + (0.941 - 0.336i)T \)
19 \( 1 + (-0.0285 + 0.999i)T \)
23 \( 1 + (-0.654 - 0.755i)T \)
29 \( 1 + (-0.564 - 0.825i)T \)
31 \( 1 + (0.198 - 0.980i)T \)
37 \( 1 + (-0.736 + 0.676i)T \)
41 \( 1 + (-0.254 - 0.967i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (0.774 - 0.633i)T \)
53 \( 1 + (0.974 - 0.226i)T \)
59 \( 1 + (-0.254 + 0.967i)T \)
61 \( 1 + (-0.998 + 0.0570i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (0.610 - 0.791i)T \)
73 \( 1 + (0.516 - 0.856i)T \)
79 \( 1 + (0.696 - 0.717i)T \)
83 \( 1 + (0.0855 - 0.996i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (-0.466 + 0.884i)T \)
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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

−29.37355358492189073864887801878, −28.1200216446809415752038409041, −27.49398393929596404042256655041, −26.407269988400736189741066414348, −25.93480026543758209003661317699, −24.987744180655276214136904254361, −23.39909869484254132867326880145, −22.179827296019340026882908472926, −21.43076274769079428121326939663, −19.79923255383193174869692187012, −19.55719129493453465394019006767, −18.30613571753712372481913483900, −17.0626824786243641155534990771, −15.93549012405795817324215561498, −15.3134020563562450709720659014, −14.34602782463115776589216640417, −12.33835011081583331842943330075, −11.109212384669620142489410833885, −10.13989433036858433583753227412, −9.40469360452941608844909312897, −8.09289048765625267722520656650, −6.95136979304364276739303141813, −5.54444188829424930228623299464, −3.46031642252854808232347601808, −2.63550794718199135822320112556, 0.4860877899932111973332169881, 2.06152942744891615360187404284, 3.628969409111079402522231583573, 5.88416760210003827725812070374, 7.17433086972673883894670399322, 7.969204224207175128689100263980, 9.10913575669881498564022676791, 10.110580578156570688340833120938, 11.98497359348427715775306363135, 12.31587557724958569508740439452, 13.73742120293107043011213494356, 15.21832705574716385476784706923, 16.645557940313763377438225947945, 16.979945648670076153976691430184, 18.622275459927405905795947382, 19.18629279896536646389917878826, 20.123455097777763837010693633901, 20.83225079452103387899852260557, 22.72967231201300441556500572585, 23.92927862804567651465764605240, 24.62999141581808267190204594543, 25.59002558666456957052417711215, 26.48216738264115048863578971849, 27.59680764879429170470580996871, 28.7410968962961407244214085894

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