↑

Properties

Label 1-4729-4729.32-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.439 + 0.898i$
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  + (0.933 − 0.358i)2-s + (0.481 + 0.876i)3-s + (0.742 − 0.669i)4-s + (0.793 + 0.608i)5-s + (0.763 + 0.645i)6-s + (0.509 + 0.860i)7-s + (0.453 − 0.891i)8-s + (−0.536 + 0.843i)9-s + (0.959 + 0.283i)10-s + (0.835 − 0.549i)11-s + (0.944 + 0.328i)12-s + (0.927 + 0.373i)13-s + (0.783 + 0.620i)14-s + (−0.150 + 0.988i)15-s + (0.103 − 0.994i)16-s + (−0.516 + 0.856i)17-s + ⋯
L(s)  = 1  + (0.933 − 0.358i)2-s + (0.481 + 0.876i)3-s + (0.742 − 0.669i)4-s + (0.793 + 0.608i)5-s + (0.763 + 0.645i)6-s + (0.509 + 0.860i)7-s + (0.453 − 0.891i)8-s + (−0.536 + 0.843i)9-s + (0.959 + 0.283i)10-s + (0.835 − 0.549i)11-s + (0.944 + 0.328i)12-s + (0.927 + 0.373i)13-s + (0.783 + 0.620i)14-s + (−0.150 + 0.988i)15-s + (0.103 − 0.994i)16-s + (−0.516 + 0.856i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.439 + 0.898i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.439 + 0.898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.950622703 + 3.087514479i\)
\(L(\frac12)\) \(\approx\) \(4.950622703 + 3.087514479i\)
\(L(1)\) \(\approx\) \(2.734322941 + 0.7735711689i\)
\(L(1)\) \(\approx\) \(2.734322941 + 0.7735711689i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.933 - 0.358i)T \)
3 \( 1 + (0.481 + 0.876i)T \)
5 \( 1 + (0.793 + 0.608i)T \)
7 \( 1 + (0.509 + 0.860i)T \)
11 \( 1 + (0.835 - 0.549i)T \)
13 \( 1 + (0.927 + 0.373i)T \)
17 \( 1 + (-0.516 + 0.856i)T \)
19 \( 1 + (0.872 + 0.488i)T \)
23 \( 1 + (0.402 + 0.915i)T \)
29 \( 1 + (0.358 - 0.933i)T \)
31 \( 1 + (-0.905 + 0.424i)T \)
37 \( 1 + (-0.894 - 0.446i)T \)
41 \( 1 + (0.417 - 0.908i)T \)
43 \( 1 + (0.956 + 0.290i)T \)
47 \( 1 + (-0.205 - 0.978i)T \)
53 \( 1 + (-0.912 + 0.410i)T \)
59 \( 1 + (-0.981 + 0.190i)T \)
61 \( 1 + (-0.657 - 0.753i)T \)
67 \( 1 + (0.991 + 0.127i)T \)
71 \( 1 + (0.984 + 0.174i)T \)
73 \( 1 + (-0.803 - 0.595i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.366 - 0.930i)T \)
89 \( 1 + (-0.726 - 0.687i)T \)
97 \( 1 + (-0.509 - 0.860i)T \)
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

−17.75740717479071138315399621924, −17.50935212499266997267220970361, −16.66089013979455568487110600008, −16.07514773635683985002178536495, −15.13197673892104730149874146591, −14.38746418131796306587062295057, −13.82853062801935902705694571545, −13.63725358540513671492123306032, −12.60259297792848132175392062172, −12.51177879093526411115014320956, −11.31820298827087795204024500240, −10.97220546593353216388640899970, −9.700487142789575379263012292801, −8.97548254811110544789946585938, −8.32521402225196963016838126765, −7.53347355327583312180212956225, −6.84641704519468831389042842113, −6.44148614886387936031358306362, −5.49398185451089991955747320249, −4.79222830841220689335741675000, −4.08608929954697910418770126547, −3.18064985204863848739070361570, −2.43254155905553341317997890081, −1.46229347663687915657896435326, −1.051952644612944056251401690723, 1.5609367807105619264344682564, 1.86510781191724115357668438547, 2.88905358793740679014846170297, 3.49429646730181534763967697976, 4.065813918486917298046662172507, 5.042697204543774283574269509994, 5.809949881392033158778472151200, 6.0316348296910669138380934480, 7.06926492397341194195549962311, 8.07887859559157425640803614352, 9.123865731776275032369818480309, 9.28721990261747433196699499167, 10.32800598448426864181788941389, 10.97246052504485506633912585784, 11.34055602953441858577551375008, 12.13757016262404089884607653323, 13.113238332438527969919172571886, 13.881657475178227121548430242354, 14.17130329378842111419569221812, 14.7384942600783036741744844770, 15.59056513065199675879382656532, 15.80371344926822152452726396258, 16.85409787730425323125320756774, 17.531456956820059793735062253704, 18.50290931185358927737991793885

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