Properties

Label 1-11e2-121.8-r1-0-0
Degree $1$
Conductor $121$
Sign $0.144 + 0.989i$
Analytic cond. $13.0032$
Root an. cond. $13.0032$
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  + (0.985 + 0.170i)2-s + (−0.809 + 0.587i)3-s + (0.941 + 0.336i)4-s + (0.993 + 0.113i)5-s + (−0.897 + 0.441i)6-s + (0.362 + 0.931i)7-s + (0.870 + 0.491i)8-s + (0.309 − 0.951i)9-s + (0.959 + 0.281i)10-s + (−0.959 + 0.281i)12-s + (0.0285 − 0.999i)13-s + (0.198 + 0.980i)14-s + (−0.870 + 0.491i)15-s + (0.774 + 0.633i)16-s + (−0.516 + 0.856i)17-s + (0.466 − 0.884i)18-s + ⋯
L(s)  = 1  + (0.985 + 0.170i)2-s + (−0.809 + 0.587i)3-s + (0.941 + 0.336i)4-s + (0.993 + 0.113i)5-s + (−0.897 + 0.441i)6-s + (0.362 + 0.931i)7-s + (0.870 + 0.491i)8-s + (0.309 − 0.951i)9-s + (0.959 + 0.281i)10-s + (−0.959 + 0.281i)12-s + (0.0285 − 0.999i)13-s + (0.198 + 0.980i)14-s + (−0.870 + 0.491i)15-s + (0.774 + 0.633i)16-s + (−0.516 + 0.856i)17-s + (0.466 − 0.884i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.144 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.144 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.144 + 0.989i$
Analytic conductor: \(13.0032\)
Root analytic conductor: \(13.0032\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (1:\ ),\ 0.144 + 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.358978798 + 2.038711083i\)
\(L(\frac12)\) \(\approx\) \(2.358978798 + 2.038711083i\)
\(L(1)\) \(\approx\) \(1.724526397 + 0.7959368482i\)
\(L(1)\) \(\approx\) \(1.724526397 + 0.7959368482i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (0.985 + 0.170i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (0.993 + 0.113i)T \)
7 \( 1 + (0.362 + 0.931i)T \)
13 \( 1 + (0.0285 - 0.999i)T \)
17 \( 1 + (-0.516 + 0.856i)T \)
19 \( 1 + (-0.0855 + 0.996i)T \)
23 \( 1 + (0.841 - 0.540i)T \)
29 \( 1 + (-0.974 + 0.226i)T \)
31 \( 1 + (-0.564 + 0.825i)T \)
37 \( 1 + (0.610 + 0.791i)T \)
41 \( 1 + (-0.696 - 0.717i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (-0.466 - 0.884i)T \)
53 \( 1 + (0.774 - 0.633i)T \)
59 \( 1 + (0.696 - 0.717i)T \)
61 \( 1 + (0.985 - 0.170i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (-0.921 - 0.389i)T \)
73 \( 1 + (0.998 + 0.0570i)T \)
79 \( 1 + (0.736 + 0.676i)T \)
83 \( 1 + (0.254 - 0.967i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (0.993 - 0.113i)T \)
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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

−28.97686651899683376582843883174, −28.03692337264804548787578316481, −26.3926021730373816911116202212, −25.129096191656469074485725664982, −24.24116048179392476307880995638, −23.557044543597062218447350672939, −22.49764815144893281144643931699, −21.621381463473342086822344378181, −20.677826567700201315361220844632, −19.487067343815040828858196329314, −18.14655861082037267747807480172, −17.04871302653702973312381212858, −16.31658303371032585639127375863, −14.63183271755654330766502838324, −13.43761195454138206099573208188, −13.18512922051844882166962207206, −11.53078290886288507040809096525, −10.958344885794264989980858111015, −9.55808432342575868080417827893, −7.32482515513500483141607598080, −6.58324206079309479333563878522, −5.33844346907997315631679410484, −4.39382136240123149391068214731, −2.33164847621867221550510570868, −1.12481332648566147783314611315, 1.871894037089749728564551724593, 3.44382349578885777407915430093, 5.0860838617087910205913258038, 5.66941466116422728483230590636, 6.679451433130704530607080707673, 8.57309912173267905796273637778, 10.173518442791114473152553762592, 11.038068111252562163087563394991, 12.30491772060792691265735524447, 13.10036843215877470930433609300, 14.695340437936429437040856474909, 15.219350394528510637618485918983, 16.54123811440246665578780218258, 17.41657992800811920200171425560, 18.4822131833185606607268761705, 20.42580903010627693833239848242, 21.228912607886067494269239201327, 22.03254946789860731980081311926, 22.625045065665035357413318021663, 23.88058151096433741581945326348, 24.90661133042344331301634121671, 25.680107878178168228724998390974, 27.06939067886610943329011239846, 28.34956358779210539916820585388, 29.04616360298823511952418010634

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