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Properties

Label 1-11e2-121.89-r0-0-0
Degree $1$
Conductor $121$
Sign $0.998 - 0.0519i$
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  + (−0.959 − 0.281i)2-s + 3-s + (0.841 + 0.540i)4-s + (−0.654 + 0.755i)5-s + (−0.959 − 0.281i)6-s + (0.415 − 0.909i)7-s + (−0.654 − 0.755i)8-s + 9-s + (0.841 − 0.540i)10-s + (0.841 + 0.540i)12-s + (0.841 + 0.540i)13-s + (−0.654 + 0.755i)14-s + (−0.654 + 0.755i)15-s + (0.415 + 0.909i)16-s + (−0.142 − 0.989i)17-s + (−0.959 − 0.281i)18-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)2-s + 3-s + (0.841 + 0.540i)4-s + (−0.654 + 0.755i)5-s + (−0.959 − 0.281i)6-s + (0.415 − 0.909i)7-s + (−0.654 − 0.755i)8-s + 9-s + (0.841 − 0.540i)10-s + (0.841 + 0.540i)12-s + (0.841 + 0.540i)13-s + (−0.654 + 0.755i)14-s + (−0.654 + 0.755i)15-s + (0.415 + 0.909i)16-s + (−0.142 − 0.989i)17-s + (−0.959 − 0.281i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.998 - 0.0519i$
Analytic conductor: \(0.561921\)
Root analytic conductor: \(0.561921\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ 0.998 - 0.0519i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9440979066 + 0.02451766684i\)
\(L(\frac12)\) \(\approx\) \(0.9440979066 + 0.02451766684i\)
\(L(1)\) \(\approx\) \(0.9347726304 - 0.03771685432i\)
\(L(1)\) \(\approx\) \(0.9347726304 - 0.03771685432i\)

Euler product

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

−28.50343872935684953042380426615, −28.02867570699452163397805233452, −27.00492272341195168595600680794, −26.08382689993840551005998359428, −25.00656904153461894596881964474, −24.484603433525250111578324842320, −23.44971017861983119884667214235, −21.462472667733011448539075691842, −20.60096955506505289807729244556, −19.71949948025149798495224251302, −18.89151934742920756898853836379, −17.89806750792712902901306308100, −16.54350207883746620381429192994, −15.3063487691451468543958734163, −15.12015877316333588810778058076, −13.30864341986035770682893850530, −12.0748893745705319878168243789, −10.79998026832072410794889252134, −9.339769921168321833209009580693, −8.43349717189469584582682697921, −8.022437074841487424545816938919, −6.37395191008884614378484471617, −4.71076531098766616999298052334, −2.9180526519906020106949742717, −1.44226520129309465437288188698, 1.53842448046720143193849152335, 3.10679138828999952442473571291, 4.03966052620333564955480460998, 6.789939974888572076468824825387, 7.57300378886281038624320836768, 8.49275247980849099428981401152, 9.77429866677448741567113544919, 10.79918548578289722115865459607, 11.772307711505545474265773449804, 13.417409868496690777236178908686, 14.506378768763366400518938499169, 15.616483481608454814414830961145, 16.58599972854949271779227008873, 18.12136045377763525398072115244, 18.79643409577453064693020674508, 19.791580986459636180986512763538, 20.55473942022929503916829680026, 21.46019414073034816040182296969, 23.086032544478102439578270745469, 24.21093296476554900652043310626, 25.44325284031294360753211697001, 26.17681881015044343660735531643, 27.09384364961998102616430704808, 27.485668522777450264168735274337, 29.194318691166121356960435760556

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