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Properties

Label 1-11e2-121.70-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  + (−0.0285 − 0.999i)2-s + (−0.809 − 0.587i)3-s + (−0.998 + 0.0570i)4-s + (0.516 + 0.856i)5-s + (−0.564 + 0.825i)6-s + (0.198 + 0.980i)7-s + (0.0855 + 0.996i)8-s + (0.309 + 0.951i)9-s + (0.841 − 0.540i)10-s + (0.841 + 0.540i)12-s + (−0.254 + 0.967i)13-s + (0.974 − 0.226i)14-s + (0.0855 − 0.996i)15-s + (0.993 − 0.113i)16-s + (−0.985 − 0.170i)17-s + (0.941 − 0.336i)18-s + ⋯
L(s)  = 1  + (−0.0285 − 0.999i)2-s + (−0.809 − 0.587i)3-s + (−0.998 + 0.0570i)4-s + (0.516 + 0.856i)5-s + (−0.564 + 0.825i)6-s + (0.198 + 0.980i)7-s + (0.0855 + 0.996i)8-s + (0.309 + 0.951i)9-s + (0.841 − 0.540i)10-s + (0.841 + 0.540i)12-s + (−0.254 + 0.967i)13-s + (0.974 − 0.226i)14-s + (0.0855 − 0.996i)15-s + (0.993 − 0.113i)16-s + (−0.985 − 0.170i)17-s + (0.941 − 0.336i)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: $\chi_{121} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ 0.998 - 0.0519i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7244336216 + 0.01881311468i\)
\(L(\frac12)\) \(\approx\) \(0.7244336216 + 0.01881311468i\)
\(L(1)\) \(\approx\) \(0.7601897087 - 0.1845849119i\)
\(L(1)\) \(\approx\) \(0.7601897087 - 0.1845849119i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (-0.0285 - 0.999i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (0.516 + 0.856i)T \)
7 \( 1 + (0.198 + 0.980i)T \)
13 \( 1 + (-0.254 + 0.967i)T \)
17 \( 1 + (-0.985 - 0.170i)T \)
19 \( 1 + (0.696 - 0.717i)T \)
23 \( 1 + (0.415 + 0.909i)T \)
29 \( 1 + (-0.466 - 0.884i)T \)
31 \( 1 + (0.774 + 0.633i)T \)
37 \( 1 + (-0.362 + 0.931i)T \)
41 \( 1 + (0.610 + 0.791i)T \)
43 \( 1 + (-0.654 - 0.755i)T \)
47 \( 1 + (0.941 + 0.336i)T \)
53 \( 1 + (0.993 + 0.113i)T \)
59 \( 1 + (0.610 - 0.791i)T \)
61 \( 1 + (-0.0285 + 0.999i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (0.897 + 0.441i)T \)
73 \( 1 + (-0.870 - 0.491i)T \)
79 \( 1 + (-0.921 - 0.389i)T \)
83 \( 1 + (-0.736 - 0.676i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (0.516 - 0.856i)T \)
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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

−28.81506560447241553200464370715, −27.857732339105264669928267077286, −26.99410791982619310767342183301, −26.1947403859638745629167708841, −24.76556938135432917923539857560, −24.13908174653977593572064155316, −23.00017636161592221479858653800, −22.29891218116563333688037347022, −21.03440022309183821325814212580, −20.06037572957900612385058961746, −18.14994251063684628251192451263, −17.37519014647072300679712486887, −16.69131390526905835480443610470, −15.83940785965424958409579834086, −14.62858628132192328817031855709, −13.37981636106402527573963024064, −12.44002954505434159483309087059, −10.66368413321431838626703430957, −9.770707762797743883345308953973, −8.57950536626836473991862520365, −7.18061322319923581488425471205, −5.888886683589688961212335607290, −4.96238096555750343599978704627, −3.98773993375463103788842888976, −0.82031345231870499430800120211, 1.754319322982875280586770339813, 2.74662441715914858723088741958, 4.74162534858445271972354231253, 5.90066121933486171248387316949, 7.19668095026387570764992184552, 8.90214018865317276061785653994, 10.03962468709834718520058479552, 11.40058269814775638436843361554, 11.70479911160817989086443525384, 13.16633924652398304368393653079, 13.967781386865573956404982742818, 15.42183270973069372455285208445, 17.1992317306883543892116936071, 17.955436174842484445200121897394, 18.7260046544100141189732235982, 19.505148938154923570467083237805, 21.226213009428459813984172406525, 21.969545462489193355576640138897, 22.57279029009529551066511503691, 23.79139868176945222603186950945, 24.9041303687988980090824968305, 26.25334172964492385165033004761, 27.26361184399266831854708761369, 28.593559980849862276616600546411, 28.829075629735160897714490692667

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