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Properties

Label 1-4729-4729.367-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.279 + 0.960i$
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  + (0.963 + 0.267i)2-s + (0.954 + 0.298i)3-s + (0.856 + 0.516i)4-s + (−0.627 + 0.778i)5-s + (0.839 + 0.543i)6-s + (0.150 + 0.988i)7-s + (0.687 + 0.726i)8-s + (0.821 + 0.569i)9-s + (−0.812 + 0.582i)10-s + (−0.0478 − 0.998i)11-s + (0.663 + 0.748i)12-s + (0.675 − 0.737i)13-s + (−0.119 + 0.992i)14-s + (−0.830 + 0.556i)15-s + (0.467 + 0.884i)16-s + (−0.127 − 0.991i)17-s + ⋯
L(s)  = 1  + (0.963 + 0.267i)2-s + (0.954 + 0.298i)3-s + (0.856 + 0.516i)4-s + (−0.627 + 0.778i)5-s + (0.839 + 0.543i)6-s + (0.150 + 0.988i)7-s + (0.687 + 0.726i)8-s + (0.821 + 0.569i)9-s + (−0.812 + 0.582i)10-s + (−0.0478 − 0.998i)11-s + (0.663 + 0.748i)12-s + (0.675 − 0.737i)13-s + (−0.119 + 0.992i)14-s + (−0.830 + 0.556i)15-s + (0.467 + 0.884i)16-s + (−0.127 − 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.279 + 0.960i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.279 + 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.147348293 + 4.192372834i\)
\(L(\frac12)\) \(\approx\) \(3.147348293 + 4.192372834i\)
\(L(1)\) \(\approx\) \(2.277665157 + 1.349256445i\)
\(L(1)\) \(\approx\) \(2.277665157 + 1.349256445i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.963 + 0.267i)T \)
3 \( 1 + (0.954 + 0.298i)T \)
5 \( 1 + (-0.627 + 0.778i)T \)
7 \( 1 + (0.150 + 0.988i)T \)
11 \( 1 + (-0.0478 - 0.998i)T \)
13 \( 1 + (0.675 - 0.737i)T \)
17 \( 1 + (-0.127 - 0.991i)T \)
19 \( 1 + (0.549 + 0.835i)T \)
23 \( 1 + (0.930 + 0.366i)T \)
29 \( 1 + (0.267 + 0.963i)T \)
31 \( 1 + (-0.936 + 0.351i)T \)
37 \( 1 + (0.887 - 0.460i)T \)
41 \( 1 + (0.595 - 0.803i)T \)
43 \( 1 + (0.620 - 0.783i)T \)
47 \( 1 + (-0.826 + 0.563i)T \)
53 \( 1 + (-0.608 + 0.793i)T \)
59 \( 1 + (-0.915 - 0.402i)T \)
61 \( 1 + (0.0159 + 0.999i)T \)
67 \( 1 + (-0.244 + 0.969i)T \)
71 \( 1 + (0.989 + 0.143i)T \)
73 \( 1 + (0.944 - 0.328i)T \)
79 \( 1 + iT \)
83 \( 1 + (0.522 - 0.852i)T \)
89 \( 1 + (0.190 - 0.981i)T \)
97 \( 1 + (-0.150 - 0.988i)T \)
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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

−18.07457848661013017661305999717, −17.123813693306033065854852587273, −16.438655319890270397831610064203, −15.75308805976174273994485000479, −15.044937705496145264732214066686, −14.65763046888736358379016900953, −13.78001628839884829214264375587, −13.14492286338507866982926170335, −12.90385403907770758035477454083, −12.093761888468845570126454502431, −11.2716063468936356000741929891, −10.75195074302630800298813696308, −9.58733726929711579815422803011, −9.31179893369235878066179576888, −7.9911521396438768983540378697, −7.79349144676704871886249488204, −6.80514816730453447948083318159, −6.41574776057850651819980193247, −4.99314548070764017936297343050, −4.47193430987182296379038479555, −3.95427439077687948998556430145, −3.32329304549580417658734964702, −2.290441442955368993376064468791, −1.50356124488456926604818049429, −0.90462591101807727278354962364, 1.303906586486773894558042305659, 2.432628005491803680225742869219, 3.07383709796442583119249954849, 3.34721390653562858012525961101, 4.167638848299794403970210975623, 5.19782503341551840364490869963, 5.68922574287821648499765500396, 6.59563415724968422152576907015, 7.48412785503698650053416588177, 7.871167737342663373752668714337, 8.68506291474916257492275509671, 9.26437269179799787533486821098, 10.51047114123428785524759909538, 11.00310862277745177255178569227, 11.59290319295839709502730715464, 12.50466079783855146105029571693, 13.01972017663259496255628877525, 14.09964793587964709344765604434, 14.169125967848378364220518136150, 14.98354379404782296139680591041, 15.62552525068819510153866023591, 15.9523470187281927582488204611, 16.52737609801868544651982601039, 17.90280554364664969905867337149, 18.49745195124283519592829363019

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