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Properties

Label 1-11e2-121.67-r0-0-0
Degree $1$
Conductor $121$
Sign $0.892 - 0.450i$
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  + (0.841 − 0.540i)2-s + 3-s + (0.415 − 0.909i)4-s + (−0.142 + 0.989i)5-s + (0.841 − 0.540i)6-s + (−0.654 + 0.755i)7-s + (−0.142 − 0.989i)8-s + 9-s + (0.415 + 0.909i)10-s + (0.415 − 0.909i)12-s + (0.415 − 0.909i)13-s + (−0.142 + 0.989i)14-s + (−0.142 + 0.989i)15-s + (−0.654 − 0.755i)16-s + (−0.959 − 0.281i)17-s + (0.841 − 0.540i)18-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)2-s + 3-s + (0.415 − 0.909i)4-s + (−0.142 + 0.989i)5-s + (0.841 − 0.540i)6-s + (−0.654 + 0.755i)7-s + (−0.142 − 0.989i)8-s + 9-s + (0.415 + 0.909i)10-s + (0.415 − 0.909i)12-s + (0.415 − 0.909i)13-s + (−0.142 + 0.989i)14-s + (−0.142 + 0.989i)15-s + (−0.654 − 0.755i)16-s + (−0.959 − 0.281i)17-s + (0.841 − 0.540i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.892 - 0.450i$
Analytic conductor: \(0.561921\)
Root analytic conductor: \(0.561921\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ 0.892 - 0.450i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.963550484 - 0.4673647071i\)
\(L(\frac12)\) \(\approx\) \(1.963550484 - 0.4673647071i\)
\(L(1)\) \(\approx\) \(1.851146532 - 0.3641541468i\)
\(L(1)\) \(\approx\) \(1.851146532 - 0.3641541468i\)

Euler product

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

−29.481570471389556437300828840176, −28.12718088433311927402774577301, −26.57252242642424974329808318120, −26.034717799102593666816101594056, −24.99072115845674285343634308774, −24.05197186003123354359129180007, −23.405871919029645148988293196214, −21.90826381883849867059580896305, −20.96571167457529935478179890081, −20.12692138416666942720681391811, −19.279493249049610243717408880867, −17.40977754482196839589777180406, −16.332512619474735750412166577153, −15.625623234715500638619357555717, −14.38917051793100121189909283528, −13.23436963753698010441811793801, −12.97223656027831398984886378524, −11.35843012476731755940385017528, −9.51972141083152244344905198870, −8.50136464553820349781787498075, −7.4129403981781401615179505076, −6.21535283548911252260237227969, −4.36976080201992622155435320397, −3.853197606562159406171606997067, −2.0836239353917443368655041692, 2.23338409626685844939198256114, 3.02635340464008995703770642635, 4.127473907468224388251382589314, 5.95877851873458357656797204605, 7.00583453961443350429246221077, 8.62812810769823624101991990162, 9.965766793095050840163281608165, 10.861252404707179492183663719782, 12.34873935639678668598758053013, 13.24632614657597472774516872991, 14.31509664841571157836938251349, 15.23652496532244966405452044508, 15.80975875189256641565989398439, 18.2280050721085033402483899064, 18.95894457341483024978833061368, 19.83381299011550398837036577847, 20.8145722892727983543893527738, 22.0101815331457734940514778397, 22.52917169695938488196149826079, 23.834553195045223633096360383190, 25.0637690817309914701173370790, 25.69920040201296585520340932158, 26.94541795695253241072831709047, 28.06650296582281583489975441268, 29.35025659355156785223151474253

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