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Properties

Label 1-229-229.21-r1-0-0
Degree $1$
Conductor $229$
Sign $0.989 - 0.144i$
Analytic cond. $24.6094$
Root an. cond. $24.6094$
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  + (0.996 + 0.0825i)2-s + (−0.677 + 0.735i)3-s + (0.986 + 0.164i)4-s + (−0.789 + 0.614i)5-s + (−0.735 + 0.677i)6-s + (0.475 − 0.879i)7-s + (0.969 + 0.245i)8-s + (−0.0825 − 0.996i)9-s + (−0.837 + 0.546i)10-s + (0.401 − 0.915i)11-s + (−0.789 + 0.614i)12-s + (−0.614 − 0.789i)13-s + (0.546 − 0.837i)14-s + (0.0825 − 0.996i)15-s + (0.945 + 0.324i)16-s + (0.789 − 0.614i)17-s + ⋯
L(s)  = 1  + (0.996 + 0.0825i)2-s + (−0.677 + 0.735i)3-s + (0.986 + 0.164i)4-s + (−0.789 + 0.614i)5-s + (−0.735 + 0.677i)6-s + (0.475 − 0.879i)7-s + (0.969 + 0.245i)8-s + (−0.0825 − 0.996i)9-s + (−0.837 + 0.546i)10-s + (0.401 − 0.915i)11-s + (−0.789 + 0.614i)12-s + (−0.614 − 0.789i)13-s + (0.546 − 0.837i)14-s + (0.0825 − 0.996i)15-s + (0.945 + 0.324i)16-s + (0.789 − 0.614i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $0.989 - 0.144i$
Analytic conductor: \(24.6094\)
Root analytic conductor: \(24.6094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (1:\ ),\ 0.989 - 0.144i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.661840490 - 0.1937808490i\)
\(L(\frac12)\) \(\approx\) \(2.661840490 - 0.1937808490i\)
\(L(1)\) \(\approx\) \(1.580723878 + 0.1600976410i\)
\(L(1)\) \(\approx\) \(1.580723878 + 0.1600976410i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (0.996 + 0.0825i)T \)
3 \( 1 + (-0.677 + 0.735i)T \)
5 \( 1 + (-0.789 + 0.614i)T \)
7 \( 1 + (0.475 - 0.879i)T \)
11 \( 1 + (0.401 - 0.915i)T \)
13 \( 1 + (-0.614 - 0.789i)T \)
17 \( 1 + (0.789 - 0.614i)T \)
19 \( 1 + (0.789 + 0.614i)T \)
23 \( 1 + (-0.837 - 0.546i)T \)
29 \( 1 + (-0.475 + 0.879i)T \)
31 \( 1 + (0.915 + 0.401i)T \)
37 \( 1 + (0.945 - 0.324i)T \)
41 \( 1 + (-0.996 - 0.0825i)T \)
43 \( 1 + (0.945 - 0.324i)T \)
47 \( 1 + (0.996 - 0.0825i)T \)
53 \( 1 + (-0.677 + 0.735i)T \)
59 \( 1 + (0.324 - 0.945i)T \)
61 \( 1 + (-0.0825 - 0.996i)T \)
67 \( 1 + (0.996 - 0.0825i)T \)
71 \( 1 + (0.401 + 0.915i)T \)
73 \( 1 + (-0.969 - 0.245i)T \)
79 \( 1 + (-0.475 - 0.879i)T \)
83 \( 1 + (0.945 + 0.324i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.245 - 0.969i)T \)
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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

−25.74594982018771552081173249434, −24.73188263805216643771731772294, −24.16504046668564724438711735870, −23.510006388742685877429692439855, −22.51988363554524581419783005296, −21.77785833550727577734461562012, −20.65536026946378566009942935951, −19.6037512474685089697444537903, −18.918880049835066187550295411113, −17.52260997286096788360809468460, −16.62311334725705064808806707023, −15.58935955859048326266371925750, −14.72518215593856499087318178855, −13.536959471612637350175523682054, −12.38838425940309513415914655092, −11.93561613850493006371087736719, −11.37228580931677977801967092591, −9.705615083054626928457871854212, −8.0160624587728400434496280317, −7.25828251860772335263454293755, −6.004087746937742647544832701944, −5.033650151919596758217884427647, −4.18420207494799001130913520858, −2.3763808594232460953313210422, −1.28819972177097725808816976873, 0.77294821609811945419540150907, 3.113086094457499785822834787863, 3.836121183882952498960208917599, 4.89259836880728032096026441821, 5.938066081784706998771087762866, 7.123141634841888543787363765302, 8.031083161512906929847535153803, 10.06707136316194390234434338514, 10.82790970767445837946239247685, 11.638960311945293371733406128649, 12.39501465120853081095277101519, 14.10225295793145925518101241773, 14.508015544991564559518345010379, 15.70946291429901027540343195936, 16.375912446223396138218204349539, 17.27157844580799675238925195330, 18.63263504583231810492712743208, 20.05657543303274501991167740676, 20.55868914960138114378529230881, 21.837080972422808487789429878120, 22.41144863963792158721988115196, 23.23304145234143867337753656120, 23.879701232328577327538240531539, 24.9099081667035721225673886248, 26.38053313351459571987047878955

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