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Properties

Label 1-4729-4729.3736-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: $\chi_{4729} (3736, \cdot )$
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

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

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