↑

Properties

Label 1-4729-4729.1013-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.462 - 0.886i$
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  + (0.305 − 0.952i)2-s + (−0.967 − 0.252i)3-s + (−0.812 − 0.582i)4-s + (−0.00797 − 0.999i)5-s + (−0.536 + 0.843i)6-s + (−0.991 + 0.127i)7-s + (−0.803 + 0.595i)8-s + (0.872 + 0.488i)9-s + (−0.954 − 0.298i)10-s + (0.645 + 0.763i)11-s + (0.639 + 0.768i)12-s + (−0.119 − 0.992i)13-s + (−0.182 + 0.983i)14-s + (−0.244 + 0.969i)15-s + (0.321 + 0.947i)16-s + (0.221 − 0.975i)17-s + ⋯
L(s)  = 1  + (0.305 − 0.952i)2-s + (−0.967 − 0.252i)3-s + (−0.812 − 0.582i)4-s + (−0.00797 − 0.999i)5-s + (−0.536 + 0.843i)6-s + (−0.991 + 0.127i)7-s + (−0.803 + 0.595i)8-s + (0.872 + 0.488i)9-s + (−0.954 − 0.298i)10-s + (0.645 + 0.763i)11-s + (0.639 + 0.768i)12-s + (−0.119 − 0.992i)13-s + (−0.182 + 0.983i)14-s + (−0.244 + 0.969i)15-s + (0.321 + 0.947i)16-s + (0.221 − 0.975i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.462 - 0.886i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1013, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.462 - 0.886i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7022501223 - 0.4259061720i\)
\(L(\frac12)\) \(\approx\) \(0.7022501223 - 0.4259061720i\)
\(L(1)\) \(\approx\) \(0.5629996677 - 0.4842158225i\)
\(L(1)\) \(\approx\) \(0.5629996677 - 0.4842158225i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.305 - 0.952i)T \)
3 \( 1 + (-0.967 - 0.252i)T \)
5 \( 1 + (-0.00797 - 0.999i)T \)
7 \( 1 + (-0.991 + 0.127i)T \)
11 \( 1 + (0.645 + 0.763i)T \)
13 \( 1 + (-0.119 - 0.992i)T \)
17 \( 1 + (0.221 - 0.975i)T \)
19 \( 1 + (0.987 - 0.158i)T \)
23 \( 1 + (0.965 - 0.260i)T \)
29 \( 1 + (-0.952 + 0.305i)T \)
31 \( 1 + (-0.852 + 0.522i)T \)
37 \( 1 + (0.742 - 0.669i)T \)
41 \( 1 + (-0.898 - 0.439i)T \)
43 \( 1 + (-0.388 + 0.921i)T \)
47 \( 1 + (-0.608 + 0.793i)T \)
53 \( 1 + (0.936 - 0.351i)T \)
59 \( 1 + (0.663 + 0.748i)T \)
61 \( 1 + (-0.726 + 0.687i)T \)
67 \( 1 + (0.0398 + 0.999i)T \)
71 \( 1 + (0.509 - 0.860i)T \)
73 \( 1 + (-0.198 - 0.980i)T \)
79 \( 1 + iT \)
83 \( 1 + (-0.213 + 0.976i)T \)
89 \( 1 + (0.328 + 0.944i)T \)
97 \( 1 + (0.991 - 0.127i)T \)
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

−18.37379126212763648256677755419, −17.149331900554696539518728556448, −16.936312744856647447856189757579, −16.37901675144909443394664560119, −15.649325711423572636819269248072, −14.99163903744757792076507854687, −14.47356010582594606966284409130, −13.49766907624248692097532360078, −13.174837244259565571450135727404, −12.127706819427235898780422591138, −11.58972287517216073499077153348, −10.93065582091377403417480184952, −9.90326063456947287850719177640, −9.571174583027333809830131455660, −8.736370962313651229944661118149, −7.603377553946534521365603422132, −6.93140462394755283945546873925, −6.546914777243666790643635673442, −5.894060367880091616261677480654, −5.361138372930283449578954198797, −4.21444408788577672701287324373, −3.61836173334181186569110189665, −3.18250987001798161540022897401, −1.65879040330249137592326096869, −0.347061060297389674163819100899, 0.75882650930073020560797028824, 1.23763337039911330992073487210, 2.28940439115138443096285644939, 3.23359459469372288824100938792, 3.99583978602072570329457899228, 4.93894071874191831623189734540, 5.28504097560304114322807420108, 5.94939299135449804762081254291, 6.94900138000300959603952883932, 7.601025167169192376248313002971, 8.79910378149361476281847257760, 9.53659807440901382834030703792, 9.7725895470576486404196977284, 10.70988121121702904976306715654, 11.45237508441437745553667808943, 12.15312471544345393502748926716, 12.4802371374077576489098241573, 13.16996574728254773914365050187, 13.481227450380150672628353247844, 14.68150851069843955422726856911, 15.37965415361660897160880014974, 16.29670840600289712386457709331, 16.63712941836049167269361890064, 17.575976238734371584480245829995, 18.047749437250155407771913279959

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