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Properties

Label 1-4729-4729.1062-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.475 + 0.879i$
Analytic cond. $21.9613$
Root an. cond. $21.9613$
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  + (−0.589 + 0.808i)2-s + (0.991 − 0.127i)3-s + (−0.305 − 0.952i)4-s + (0.964 + 0.262i)5-s + (−0.481 + 0.876i)6-s + (−0.443 − 0.896i)7-s + (0.949 + 0.313i)8-s + (0.967 − 0.252i)9-s + (−0.780 + 0.624i)10-s + (−0.939 − 0.343i)11-s + (−0.424 − 0.905i)12-s + (−0.316 + 0.948i)13-s + (0.985 + 0.169i)14-s + (0.990 + 0.137i)15-s + (−0.812 + 0.582i)16-s + (−0.593 − 0.804i)17-s + ⋯
L(s)  = 1  + (−0.589 + 0.808i)2-s + (0.991 − 0.127i)3-s + (−0.305 − 0.952i)4-s + (0.964 + 0.262i)5-s + (−0.481 + 0.876i)6-s + (−0.443 − 0.896i)7-s + (0.949 + 0.313i)8-s + (0.967 − 0.252i)9-s + (−0.780 + 0.624i)10-s + (−0.939 − 0.343i)11-s + (−0.424 − 0.905i)12-s + (−0.316 + 0.948i)13-s + (0.985 + 0.169i)14-s + (0.990 + 0.137i)15-s + (−0.812 + 0.582i)16-s + (−0.593 − 0.804i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.475 + 0.879i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1062, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.475 + 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.746131203 + 1.041309731i\)
\(L(\frac12)\) \(\approx\) \(1.746131203 + 1.041309731i\)
\(L(1)\) \(\approx\) \(1.163181118 + 0.3499561202i\)
\(L(1)\) \(\approx\) \(1.163181118 + 0.3499561202i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.589 + 0.808i)T \)
3 \( 1 + (0.991 - 0.127i)T \)
5 \( 1 + (0.964 + 0.262i)T \)
7 \( 1 + (-0.443 - 0.896i)T \)
11 \( 1 + (-0.939 - 0.343i)T \)
13 \( 1 + (-0.316 + 0.948i)T \)
17 \( 1 + (-0.593 - 0.804i)T \)
19 \( 1 + (0.429 + 0.903i)T \)
23 \( 1 + (0.793 - 0.608i)T \)
29 \( 1 + (0.589 + 0.808i)T \)
31 \( 1 + (-0.872 + 0.488i)T \)
37 \( 1 + (0.156 + 0.987i)T \)
41 \( 1 + (0.848 + 0.529i)T \)
43 \( 1 + (0.947 + 0.318i)T \)
47 \( 1 + (-0.659 + 0.751i)T \)
53 \( 1 + (-0.0823 - 0.996i)T \)
59 \( 1 + (-0.994 - 0.100i)T \)
61 \( 1 + (-0.993 - 0.116i)T \)
67 \( 1 + (0.239 + 0.970i)T \)
71 \( 1 + (0.504 + 0.863i)T \)
73 \( 1 + (0.935 + 0.353i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.361 + 0.932i)T \)
89 \( 1 + (0.937 - 0.348i)T \)
97 \( 1 + (-0.443 - 0.896i)T \)
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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

−18.142239662652263780312626651306, −17.64025663090839281019802011984, −16.93158579272725497110464486736, −15.87410915117183232980314784291, −15.47828340975946252035666040672, −14.76501270713022323774453349078, −13.60668862482255776988326997921, −13.31918234378067628584584288068, −12.64573949647658035484927480923, −12.29623222361008984029376769182, −10.85164875327903728048080332043, −10.589491950238700408334487937108, −9.59910554295549786265477713089, −9.31422390893441752730025852495, −8.78063743583828511932666235372, −7.89224672862792711405235676750, −7.40906801557620566198937234666, −6.30906129765974636577762618883, −5.319160577038308984309421007793, −4.71971963590907659766262369922, −3.64032256782567479016669160135, −2.81555603978798682733866743699, −2.39326959017879590683373889823, −1.8512855560275624246175261610, −0.64920252148576051893774861380, 0.93364048363987070140059639125, 1.68276046666873643909959919377, 2.59458962951851297122633698613, 3.27749105013703742637924446018, 4.48685851954493723378084164726, 5.00210731057196659751198433918, 6.09562890523956860673846774782, 6.77804250032567065820675392177, 7.202271679407931763633253173041, 7.93474430045092425691916371553, 8.734217257916057199722858055159, 9.46780706728734535181491051491, 9.76452122940872015172291927414, 10.58005040230539603258944589384, 11.07950527309218027726068155428, 12.62624606335135704778912812711, 13.182026318250022095616973158999, 13.87965238705628336749406739696, 14.222260642525515771783441363852, 14.73737232725360404989547619096, 15.77494703009189462434458656701, 16.30442011070761567149377367846, 16.76939105620613929331790676416, 17.75538017989770625777558814940, 18.26518792877496494675402252953

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