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Properties

Label 1-229-229.86-r1-0-0
Degree $1$
Conductor $229$
Sign $0.998 + 0.0581i$
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  + (0.324 + 0.945i)2-s + (−0.986 − 0.164i)3-s + (−0.789 + 0.614i)4-s + (0.879 − 0.475i)5-s + (−0.164 − 0.986i)6-s + (−0.915 + 0.401i)7-s + (−0.837 − 0.546i)8-s + (0.945 + 0.324i)9-s + (0.735 + 0.677i)10-s + (0.0825 + 0.996i)11-s + (0.879 − 0.475i)12-s + (−0.475 − 0.879i)13-s + (−0.677 − 0.735i)14-s + (−0.945 + 0.324i)15-s + (0.245 − 0.969i)16-s + (−0.879 + 0.475i)17-s + ⋯
L(s)  = 1  + (0.324 + 0.945i)2-s + (−0.986 − 0.164i)3-s + (−0.789 + 0.614i)4-s + (0.879 − 0.475i)5-s + (−0.164 − 0.986i)6-s + (−0.915 + 0.401i)7-s + (−0.837 − 0.546i)8-s + (0.945 + 0.324i)9-s + (0.735 + 0.677i)10-s + (0.0825 + 0.996i)11-s + (0.879 − 0.475i)12-s + (−0.475 − 0.879i)13-s + (−0.677 − 0.735i)14-s + (−0.945 + 0.324i)15-s + (0.245 − 0.969i)16-s + (−0.879 + 0.475i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $0.998 + 0.0581i$
Analytic conductor: \(24.6094\)
Root analytic conductor: \(24.6094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (1:\ ),\ 0.998 + 0.0581i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.086424542 + 0.03162034083i\)
\(L(\frac12)\) \(\approx\) \(1.086424542 + 0.03162034083i\)
\(L(1)\) \(\approx\) \(0.7959878735 + 0.2827342660i\)
\(L(1)\) \(\approx\) \(0.7959878735 + 0.2827342660i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (0.324 + 0.945i)T \)
3 \( 1 + (-0.986 - 0.164i)T \)
5 \( 1 + (0.879 - 0.475i)T \)
7 \( 1 + (-0.915 + 0.401i)T \)
11 \( 1 + (0.0825 + 0.996i)T \)
13 \( 1 + (-0.475 - 0.879i)T \)
17 \( 1 + (-0.879 + 0.475i)T \)
19 \( 1 + (-0.879 - 0.475i)T \)
23 \( 1 + (0.735 - 0.677i)T \)
29 \( 1 + (0.915 - 0.401i)T \)
31 \( 1 + (0.996 - 0.0825i)T \)
37 \( 1 + (0.245 + 0.969i)T \)
41 \( 1 + (-0.324 - 0.945i)T \)
43 \( 1 + (0.245 + 0.969i)T \)
47 \( 1 + (0.324 - 0.945i)T \)
53 \( 1 + (-0.986 - 0.164i)T \)
59 \( 1 + (0.969 + 0.245i)T \)
61 \( 1 + (0.945 + 0.324i)T \)
67 \( 1 + (0.324 - 0.945i)T \)
71 \( 1 + (0.0825 - 0.996i)T \)
73 \( 1 + (0.837 + 0.546i)T \)
79 \( 1 + (0.915 + 0.401i)T \)
83 \( 1 + (0.245 - 0.969i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.546 - 0.837i)T \)
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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

−26.6918260754485552310625872624, −25.119584170409691381266574558056, −23.87350437783755932425811689160, −23.09739612023082452254677936362, −22.17633690573720603571679802743, −21.687659032953283115450307463703, −20.88768062986052764787496723842, −19.35833675663603858283176884779, −18.83888013737421068161072885715, −17.67528216758089601013391585413, −16.92669679784899695065855944300, −15.779717793877570647424596367261, −14.33518702179313899004828499396, −13.47172641814503774018819812460, −12.66117437288635891661047121240, −11.46152243315879175407600562058, −10.7181335496201030635653273521, −9.86678413729683593903348978413, −9.04942051033023440756031633126, −6.75311231804302075162386718117, −6.1264074532866542105079097927, −4.94306193014106329760005369330, −3.74870808243042332591350980424, −2.46067017965072627140329543605, −0.9506148338464927493213372287, 0.46461730498994548261281192738, 2.47089928087530322723042926960, 4.447778584971324925676036760590, 5.19625287735897921952347552761, 6.35471011435505049845183952, 6.767727149058991568533493557144, 8.36169871791076846962337659933, 9.55237441242995642263082538623, 10.363679264146155542877408425615, 12.20493343559026342404095095312, 12.80953027223875070911102879119, 13.41815435854838911137075630223, 15.04318671864093059323659251287, 15.71053934406173973927315993393, 16.89604708138910576569621188283, 17.39312988514078551099358643642, 18.12356218832977994310345935562, 19.3900948888659871561471317302, 20.91827117421372217013284865920, 21.97930091943721006051059818241, 22.47478370852127858259424865021, 23.31836303714018821745013953477, 24.412997753437167435544265647098, 25.09638950916356930520361308312, 25.795773048405265408113305427141

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