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Properties

Label 1-11e2-121.78-r0-0-0
Degree $1$
Conductor $121$
Sign $0.167 - 0.985i$
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  + (−0.142 − 0.989i)2-s + 3-s + (−0.959 + 0.281i)4-s + (0.415 − 0.909i)5-s + (−0.142 − 0.989i)6-s + (0.841 + 0.540i)7-s + (0.415 + 0.909i)8-s + 9-s + (−0.959 − 0.281i)10-s + (−0.959 + 0.281i)12-s + (−0.959 + 0.281i)13-s + (0.415 − 0.909i)14-s + (0.415 − 0.909i)15-s + (0.841 − 0.540i)16-s + (−0.654 − 0.755i)17-s + (−0.142 − 0.989i)18-s + ⋯
L(s)  = 1  + (−0.142 − 0.989i)2-s + 3-s + (−0.959 + 0.281i)4-s + (0.415 − 0.909i)5-s + (−0.142 − 0.989i)6-s + (0.841 + 0.540i)7-s + (0.415 + 0.909i)8-s + 9-s + (−0.959 − 0.281i)10-s + (−0.959 + 0.281i)12-s + (−0.959 + 0.281i)13-s + (0.415 − 0.909i)14-s + (0.415 − 0.909i)15-s + (0.841 − 0.540i)16-s + (−0.654 − 0.755i)17-s + (−0.142 − 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.167 - 0.985i$
Analytic conductor: \(0.561921\)
Root analytic conductor: \(0.561921\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (78, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ 0.167 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.032784937 - 0.8716989903i\)
\(L(\frac12)\) \(\approx\) \(1.032784937 - 0.8716989903i\)
\(L(1)\) \(\approx\) \(1.126094028 - 0.6452013010i\)
\(L(1)\) \(\approx\) \(1.126094028 - 0.6452013010i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (-0.142 - 0.989i)T \)
3 \( 1 + T \)
5 \( 1 + (0.415 - 0.909i)T \)
7 \( 1 + (0.841 + 0.540i)T \)
13 \( 1 + (-0.959 + 0.281i)T \)
17 \( 1 + (-0.654 - 0.755i)T \)
19 \( 1 + (-0.654 + 0.755i)T \)
23 \( 1 + (0.841 - 0.540i)T \)
29 \( 1 + (-0.654 + 0.755i)T \)
31 \( 1 + (-0.959 - 0.281i)T \)
37 \( 1 + (-0.959 - 0.281i)T \)
41 \( 1 + (-0.142 - 0.989i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 + (-0.142 + 0.989i)T \)
53 \( 1 + (0.841 + 0.540i)T \)
59 \( 1 + (-0.142 + 0.989i)T \)
61 \( 1 + (-0.142 + 0.989i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (0.841 - 0.540i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (0.841 + 0.540i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (0.415 + 0.909i)T \)
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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

−29.539775669016822238128108281864, −27.75860313776905885848662302118, −26.80136288796443923633020942161, −26.30288168867190627639711033743, −25.318256802159042505696000487280, −24.42793201154812744377887310965, −23.531385143358696769363114879029, −22.14442074130182850655071491465, −21.332159728827769738446398162297, −19.82336924888089090183724561120, −18.92192545688089483512948904196, −17.784735216816669197839648675850, −17.05826450825711454368430020494, −15.215854310352055371311792395844, −14.9026766947609524555010547828, −13.87569585919997066693480138953, −13.04125692436729441232171979367, −10.84611739583321243609603405943, −9.81754463774178280153877077385, −8.642340569197466749958472774899, −7.50644551917393975593997073509, −6.784151622314406067543126854628, −5.05704445132779807638334315113, −3.73241860775738162961535700604, −2.00981796062759923910593522062, 1.63159705065535801866670661216, 2.52559411147648372920380535046, 4.24896360787179047444210741626, 5.17744817483070916814214456410, 7.59110784699413844718271381286, 8.83081143980503652236482122090, 9.24042709786215529026500179094, 10.628085847445509555716924238071, 12.112650791449302014748424464147, 12.88840850522499564658041195477, 14.02740430921745825429511231483, 14.891263596845261217616235283909, 16.58139687413650520714575942164, 17.76846732752040175840756355379, 18.76216723308723394414207452058, 19.80418329814039226466480180270, 20.74298452101753975652298219285, 21.226739968768623405054388584150, 22.29539026657996430618889496277, 24.04725682184479648390365322990, 24.77038231242903416712709181509, 25.89309748300532270870683858569, 27.1747496523275370766807644778, 27.66070782875308747111372230501, 29.00383811551089543133510576065

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