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Properties

Label 1-11e2-121.14-r0-0-0
Degree $1$
Conductor $121$
Sign $-0.885 + 0.464i$
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  + (0.897 + 0.441i)2-s + (−0.809 + 0.587i)3-s + (0.610 + 0.791i)4-s + (−0.736 + 0.676i)5-s + (−0.985 + 0.170i)6-s + (−0.998 − 0.0570i)7-s + (0.198 + 0.980i)8-s + (0.309 − 0.951i)9-s + (−0.959 + 0.281i)10-s + (−0.959 − 0.281i)12-s + (−0.564 + 0.825i)13-s + (−0.870 − 0.491i)14-s + (0.198 − 0.980i)15-s + (−0.254 + 0.967i)16-s + (−0.921 − 0.389i)17-s + (0.696 − 0.717i)18-s + ⋯
L(s)  = 1  + (0.897 + 0.441i)2-s + (−0.809 + 0.587i)3-s + (0.610 + 0.791i)4-s + (−0.736 + 0.676i)5-s + (−0.985 + 0.170i)6-s + (−0.998 − 0.0570i)7-s + (0.198 + 0.980i)8-s + (0.309 − 0.951i)9-s + (−0.959 + 0.281i)10-s + (−0.959 − 0.281i)12-s + (−0.564 + 0.825i)13-s + (−0.870 − 0.491i)14-s + (0.198 − 0.980i)15-s + (−0.254 + 0.967i)16-s + (−0.921 − 0.389i)17-s + (0.696 − 0.717i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $-0.885 + 0.464i$
Analytic conductor: \(0.561921\)
Root analytic conductor: \(0.561921\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ -0.885 + 0.464i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2262759857 + 0.9188274751i\)
\(L(\frac12)\) \(\approx\) \(0.2262759857 + 0.9188274751i\)
\(L(1)\) \(\approx\) \(0.7642515785 + 0.6916035976i\)
\(L(1)\) \(\approx\) \(0.7642515785 + 0.6916035976i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (0.897 + 0.441i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.736 + 0.676i)T \)
7 \( 1 + (-0.998 - 0.0570i)T \)
13 \( 1 + (-0.564 + 0.825i)T \)
17 \( 1 + (-0.921 - 0.389i)T \)
19 \( 1 + (0.974 + 0.226i)T \)
23 \( 1 + (0.841 + 0.540i)T \)
29 \( 1 + (0.0855 + 0.996i)T \)
31 \( 1 + (-0.0285 + 0.999i)T \)
37 \( 1 + (0.941 + 0.336i)T \)
41 \( 1 + (-0.466 - 0.884i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 + (0.696 + 0.717i)T \)
53 \( 1 + (-0.254 - 0.967i)T \)
59 \( 1 + (-0.466 + 0.884i)T \)
61 \( 1 + (0.897 - 0.441i)T \)
67 \( 1 + (-0.142 + 0.989i)T \)
71 \( 1 + (0.516 - 0.856i)T \)
73 \( 1 + (-0.362 - 0.931i)T \)
79 \( 1 + (0.993 - 0.113i)T \)
83 \( 1 + (0.774 + 0.633i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (-0.736 - 0.676i)T \)
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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

−28.667068855721634431395158503856, −28.21792842532882445679102296040, −26.841359185930289777623549033506, −24.946778750951052928964664760040, −24.45757490336038356825071918805, −23.31995244180138620016755850844, −22.67223086794427608356848090958, −21.865854917630507428976170707344, −20.276995734249080533733449543962, −19.59732022691600744794074368951, −18.63684196343928830691081888912, −17.06282320884958352401575168404, −16.02970963004695287230160552687, −15.18421027511061503588892296041, −13.33046068171785149187036051422, −12.82774289082870371557904748934, −11.91025854749127079305593290553, −10.94857353474449480422158660257, −9.60586052069537535346348387770, −7.69089319011884400674259952096, −6.51351295755044763938665245592, −5.384562879195695334718981009207, −4.258451968072923654449063027715, −2.67846890623578467324239256557, −0.73506328149887974957117021699, 3.005523386719594284350523844353, 4.01410505154144700137617994056, 5.20831581288316969454298615784, 6.62925695731973513168525435643, 7.17918513846010950808774303168, 9.19212370162404441240785948930, 10.6697986659018041073947611754, 11.6537308744785520141525266062, 12.48004049674675096784594192932, 13.91155896175783861197024350251, 15.12396893288243785330282030204, 15.92255766795029089580420445926, 16.60825185612323380103731916457, 17.91523097919121151884634345549, 19.35704445093847266231589144133, 20.52416652006584327211785269110, 22.01385741607515204217749137483, 22.26864260221097362469755064169, 23.29164443056725056712854018682, 23.992235307727595377101563221022, 25.45980984474875890368781486145, 26.58250044330944581111611544278, 27.09027522941628443889819968096, 28.913263183009468515345209838958, 29.26599219105291046205109729835

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