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Properties

Label 1-11e2-121.15-r0-0-0
Degree $1$
Conductor $121$
Sign $0.989 + 0.144i$
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  + (−0.985 + 0.170i)2-s + (−0.809 − 0.587i)3-s + (0.941 − 0.336i)4-s + (0.993 − 0.113i)5-s + (0.897 + 0.441i)6-s + (−0.362 + 0.931i)7-s + (−0.870 + 0.491i)8-s + (0.309 + 0.951i)9-s + (−0.959 + 0.281i)10-s + (−0.959 − 0.281i)12-s + (−0.0285 − 0.999i)13-s + (0.198 − 0.980i)14-s + (−0.870 − 0.491i)15-s + (0.774 − 0.633i)16-s + (0.516 + 0.856i)17-s + (−0.466 − 0.884i)18-s + ⋯
L(s)  = 1  + (−0.985 + 0.170i)2-s + (−0.809 − 0.587i)3-s + (0.941 − 0.336i)4-s + (0.993 − 0.113i)5-s + (0.897 + 0.441i)6-s + (−0.362 + 0.931i)7-s + (−0.870 + 0.491i)8-s + (0.309 + 0.951i)9-s + (−0.959 + 0.281i)10-s + (−0.959 − 0.281i)12-s + (−0.0285 − 0.999i)13-s + (0.198 − 0.980i)14-s + (−0.870 − 0.491i)15-s + (0.774 − 0.633i)16-s + (0.516 + 0.856i)17-s + (−0.466 − 0.884i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.989 + 0.144i$
Analytic conductor: \(0.561921\)
Root analytic conductor: \(0.561921\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ 0.989 + 0.144i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6196461659 + 0.04512657031i\)
\(L(\frac12)\) \(\approx\) \(0.6196461659 + 0.04512657031i\)
\(L(1)\) \(\approx\) \(0.6525091186 + 0.005524492688i\)
\(L(1)\) \(\approx\) \(0.6525091186 + 0.005524492688i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (-0.985 + 0.170i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (0.993 - 0.113i)T \)
7 \( 1 + (-0.362 + 0.931i)T \)
13 \( 1 + (-0.0285 - 0.999i)T \)
17 \( 1 + (0.516 + 0.856i)T \)
19 \( 1 + (0.0855 + 0.996i)T \)
23 \( 1 + (0.841 + 0.540i)T \)
29 \( 1 + (0.974 + 0.226i)T \)
31 \( 1 + (-0.564 - 0.825i)T \)
37 \( 1 + (0.610 - 0.791i)T \)
41 \( 1 + (0.696 - 0.717i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 + (-0.466 + 0.884i)T \)
53 \( 1 + (0.774 + 0.633i)T \)
59 \( 1 + (0.696 + 0.717i)T \)
61 \( 1 + (-0.985 - 0.170i)T \)
67 \( 1 + (-0.142 + 0.989i)T \)
71 \( 1 + (-0.921 + 0.389i)T \)
73 \( 1 + (-0.998 + 0.0570i)T \)
79 \( 1 + (-0.736 + 0.676i)T \)
83 \( 1 + (-0.254 - 0.967i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (0.993 + 0.113i)T \)
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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

−29.08390189755602852105985334967, −28.120553911938897004396169264131, −26.88859532502499259992504212655, −26.360533363620605955820618999799, −25.312351955974780281619413111118, −24.03915355017233480406294678524, −22.85895240026599435754766156335, −21.59433109478307628936243794277, −20.98330647274257776847334669884, −19.821117183808058000769102587451, −18.45823900169702600720398429693, −17.583257424198791156883297133, −16.71995615101163018332653990653, −16.12111372709790259470325324913, −14.54617028376608200168435898573, −13.10349931519920097000048496468, −11.67626841439741620163174713222, −10.70210293427934664554105865241, −9.8328453993645610122484024294, −9.07926477173334210164618087003, −7.06724694297071514100949687460, −6.37486528616251066564365604701, −4.75119443023947297672747074812, −2.98180940970223898431143480769, −1.08007999276501528129520498853, 1.32648672429399046783191691320, 2.6099452721686794090759419198, 5.67427503692102616485324851095, 5.892987033833711545973918257788, 7.37910834143777172330903183486, 8.63776348856496726815711974779, 9.8958961818261578304448010216, 10.762731994531565604477545906045, 12.16771552361853344414360215235, 12.95978361370054696130771461518, 14.65290259438703141524324422386, 15.9447277909552507969650959960, 16.97226725413492340663259925523, 17.751016911921741753396563416031, 18.55117988859630046731124128580, 19.428234875223535389843436668508, 20.92381973674539370914391686658, 21.906559343535864719169966416159, 23.103536210058832759769865181528, 24.41912080910700763233911036880, 25.16018167888141820232983575264, 25.72482726363050921556664792089, 27.38855351446612230892474871683, 28.11452070022652460177921354723, 29.07085123327091425217126062107

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