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Properties

Label 1-11e2-121.75-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  + (0.610 + 0.791i)2-s + (0.309 − 0.951i)3-s + (−0.254 + 0.967i)4-s + (0.0855 − 0.996i)5-s + (0.941 − 0.336i)6-s + (0.993 + 0.113i)7-s + (−0.921 + 0.389i)8-s + (−0.809 − 0.587i)9-s + (0.841 − 0.540i)10-s + (0.841 + 0.540i)12-s + (−0.362 − 0.931i)13-s + (0.516 + 0.856i)14-s + (−0.921 − 0.389i)15-s + (−0.870 − 0.491i)16-s + (0.696 + 0.717i)17-s + (−0.0285 − 0.999i)18-s + ⋯
L(s)  = 1  + (0.610 + 0.791i)2-s + (0.309 − 0.951i)3-s + (−0.254 + 0.967i)4-s + (0.0855 − 0.996i)5-s + (0.941 − 0.336i)6-s + (0.993 + 0.113i)7-s + (−0.921 + 0.389i)8-s + (−0.809 − 0.587i)9-s + (0.841 − 0.540i)10-s + (0.841 + 0.540i)12-s + (−0.362 − 0.931i)13-s + (0.516 + 0.856i)14-s + (−0.921 − 0.389i)15-s + (−0.870 − 0.491i)16-s + (0.696 + 0.717i)17-s + (−0.0285 − 0.999i)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: $\chi_{121} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ 0.998 - 0.0519i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.576339639 - 0.04093661245i\)
\(L(\frac12)\) \(\approx\) \(1.576339639 - 0.04093661245i\)
\(L(1)\) \(\approx\) \(1.505787210 + 0.05706635970i\)
\(L(1)\) \(\approx\) \(1.505787210 + 0.05706635970i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (0.610 + 0.791i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.0855 - 0.996i)T \)
7 \( 1 + (0.993 + 0.113i)T \)
13 \( 1 + (-0.362 - 0.931i)T \)
17 \( 1 + (0.696 + 0.717i)T \)
19 \( 1 + (0.897 + 0.441i)T \)
23 \( 1 + (0.415 + 0.909i)T \)
29 \( 1 + (-0.985 + 0.170i)T \)
31 \( 1 + (-0.998 - 0.0570i)T \)
37 \( 1 + (0.774 + 0.633i)T \)
41 \( 1 + (-0.564 + 0.825i)T \)
43 \( 1 + (-0.654 - 0.755i)T \)
47 \( 1 + (-0.0285 + 0.999i)T \)
53 \( 1 + (-0.870 + 0.491i)T \)
59 \( 1 + (-0.564 - 0.825i)T \)
61 \( 1 + (0.610 - 0.791i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (-0.466 - 0.884i)T \)
73 \( 1 + (-0.736 + 0.676i)T \)
79 \( 1 + (0.974 - 0.226i)T \)
83 \( 1 + (0.198 + 0.980i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (0.0855 + 0.996i)T \)
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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

−29.12858267013320291373796369466, −28.04846530642655336339866355105, −27.04474423140172737771222095733, −26.47177838341327339808807452332, −24.96858600201543342294809159942, −23.71580973771822931015766047024, −22.600032221941871427082833394791, −21.854024577290758143280485139480, −21.02163367989371922767298977529, −20.19114289850739826398055410950, −18.96662906887892679489014607817, −18.01963743305454898096216531605, −16.4442710901667007698249183759, −14.96617618197063600540090813318, −14.487791280128315757969209478042, −13.65312418744085550180504664946, −11.72052269275314443316547994610, −11.1047543322197983612228763753, −10.07117455129095641580940991261, −9.07200334259332479058159672842, −7.29429024836635434880511649538, −5.54125841384088571226928981030, −4.49188512217534278350106778660, −3.2903496037499023811903241064, −2.12340012471946204735515207825, 1.553842321826083923970790498962, 3.38815446483930034391872210005, 5.07838362948115743298987245182, 5.84813783791406253857601927771, 7.62787870040699411492556102489, 8.03487619440832657569753669839, 9.266490870961088560018866646, 11.565797596339195713612145062293, 12.51159327734430303926961715705, 13.28414291735956270753677511449, 14.38025812701428784289814505274, 15.26068303571316546529579483316, 16.78133238099622933318819735989, 17.49431791445523495462307551630, 18.47070955064189101579560544282, 20.1191837268155743077487917582, 20.81880646071014369760876826158, 22.05617938425468175647637970334, 23.52317475907305873468067455001, 23.97149872011256789957381135107, 24.98253665318033064180094295170, 25.39523685395524285872905291513, 26.907255824747221105630865212728, 27.93752015822259656613758279645, 29.350371971937073922470149441052

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