↑

Properties

Label 1-4729-4729.131-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.363 + 0.931i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.495 + 0.868i)2-s + (−0.812 − 0.582i)3-s + (−0.509 + 0.860i)4-s + (−0.0132 + 0.999i)5-s + (0.103 − 0.994i)6-s + (0.977 + 0.211i)7-s + (−0.999 − 0.0159i)8-s + (0.321 + 0.947i)9-s + (−0.875 + 0.483i)10-s + (0.601 − 0.798i)11-s + (0.915 − 0.402i)12-s + (0.749 + 0.661i)13-s + (0.300 + 0.953i)14-s + (0.593 − 0.804i)15-s + (−0.481 − 0.876i)16-s + (−0.780 − 0.624i)17-s + ⋯
L(s)  = 1  + (0.495 + 0.868i)2-s + (−0.812 − 0.582i)3-s + (−0.509 + 0.860i)4-s + (−0.0132 + 0.999i)5-s + (0.103 − 0.994i)6-s + (0.977 + 0.211i)7-s + (−0.999 − 0.0159i)8-s + (0.321 + 0.947i)9-s + (−0.875 + 0.483i)10-s + (0.601 − 0.798i)11-s + (0.915 − 0.402i)12-s + (0.749 + 0.661i)13-s + (0.300 + 0.953i)14-s + (0.593 − 0.804i)15-s + (−0.481 − 0.876i)16-s + (−0.780 − 0.624i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.363 + 0.931i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.363 + 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.480248834 + 1.011189465i\)
\(L(\frac12)\) \(\approx\) \(1.480248834 + 1.011189465i\)
\(L(1)\) \(\approx\) \(1.008751047 + 0.5259245960i\)
\(L(1)\) \(\approx\) \(1.008751047 + 0.5259245960i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.495 + 0.868i)T \)
3 \( 1 + (-0.812 - 0.582i)T \)
5 \( 1 + (-0.0132 + 0.999i)T \)
7 \( 1 + (0.977 + 0.211i)T \)
11 \( 1 + (0.601 - 0.798i)T \)
13 \( 1 + (0.749 + 0.661i)T \)
17 \( 1 + (-0.780 - 0.624i)T \)
19 \( 1 + (-0.254 - 0.966i)T \)
23 \( 1 + (-0.424 + 0.905i)T \)
29 \( 1 + (0.495 - 0.868i)T \)
31 \( 1 + (-0.793 + 0.608i)T \)
37 \( 1 + (0.643 - 0.765i)T \)
41 \( 1 + (-0.687 - 0.726i)T \)
43 \( 1 + (-0.140 - 0.990i)T \)
47 \( 1 + (0.999 - 0.0425i)T \)
53 \( 1 + (0.997 + 0.0743i)T \)
59 \( 1 + (-0.161 + 0.986i)T \)
61 \( 1 + (0.952 - 0.303i)T \)
67 \( 1 + (-0.0663 + 0.997i)T \)
71 \( 1 + (0.777 + 0.629i)T \)
73 \( 1 + (-0.326 + 0.945i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.635 - 0.772i)T \)
89 \( 1 + (0.882 - 0.469i)T \)
97 \( 1 + (0.977 + 0.211i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.09584970689310575921385143770, −17.385230263630041582221609468675, −16.79244555980422154793970403639, −16.06803021954944025875512826199, −15.040675091536572970544984753000, −14.90529519629017996279026853027, −13.92213281218080059539864942119, −13.03379166396137409369581466177, −12.52590940066787188492857589974, −11.96932526487943244192983530546, −11.30227539004256603334344958207, −10.67189228603443287778069762277, −10.115374440328331457897174423579, −9.34637057288162523044396609328, −8.61313554323448709596574867872, −8.00434425604610274215144853365, −6.598355071804171515662955232634, −5.99687349961282004866952900436, −5.22007807810500692205563498427, −4.62259401272820669380470314809, −4.14399481545992435264484354932, −3.55478265603253312141841002442, −2.09671169552606578783956360282, −1.43889016495580286090640630347, −0.76452677066641369993341587133, 0.64990873194932613453942053512, 1.92779769990002218496083637586, 2.62126336811539109737942164572, 3.82372991908360901514387317588, 4.31715442111332047878089819589, 5.37537051792751350509069850653, 5.80684709682516676059106920202, 6.60362456372381971286471326746, 7.06738670709152479909577231380, 7.6535220385022641009586361446, 8.62321312361461075562716135864, 9.05711746465886668335523964738, 10.37478826764852439597414832168, 11.241203619229292447234281993245, 11.52515019836065152561753117677, 12.02603962663986556227610755492, 13.232600381131812630202810031069, 13.741659578976093895230484926264, 14.092316710359272182800101217646, 14.93757360267205613643520328156, 15.73210190542472547818675092814, 16.10388382192692650555229623536, 17.1658244644675697902375313553, 17.51270643758423256406813226626, 18.12909711346791221252790612774

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