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Properties

Label 1-4729-4729.607-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.997 - 0.0676i$
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  + (−0.921 − 0.388i)2-s + (0.887 − 0.460i)3-s + (0.698 + 0.715i)4-s + (0.839 − 0.543i)5-s + (−0.996 + 0.0796i)6-s + (−0.971 − 0.236i)7-s + (−0.366 − 0.930i)8-s + (0.576 − 0.817i)9-s + (−0.984 + 0.174i)10-s + (−0.967 + 0.252i)11-s + (0.949 + 0.313i)12-s + (−0.687 − 0.726i)13-s + (0.803 + 0.595i)14-s + (0.495 − 0.868i)15-s + (−0.0239 + 0.999i)16-s + (0.933 + 0.358i)17-s + ⋯
L(s)  = 1  + (−0.921 − 0.388i)2-s + (0.887 − 0.460i)3-s + (0.698 + 0.715i)4-s + (0.839 − 0.543i)5-s + (−0.996 + 0.0796i)6-s + (−0.971 − 0.236i)7-s + (−0.366 − 0.930i)8-s + (0.576 − 0.817i)9-s + (−0.984 + 0.174i)10-s + (−0.967 + 0.252i)11-s + (0.949 + 0.313i)12-s + (−0.687 − 0.726i)13-s + (0.803 + 0.595i)14-s + (0.495 − 0.868i)15-s + (−0.0239 + 0.999i)16-s + (0.933 + 0.358i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.997 - 0.0676i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.997 - 0.0676i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03561144417 - 1.051638777i\)
\(L(\frac12)\) \(\approx\) \(0.03561144417 - 1.051638777i\)
\(L(1)\) \(\approx\) \(0.7573659720 - 0.4626006107i\)
\(L(1)\) \(\approx\) \(0.7573659720 - 0.4626006107i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.921 - 0.388i)T \)
3 \( 1 + (0.887 - 0.460i)T \)
5 \( 1 + (0.839 - 0.543i)T \)
7 \( 1 + (-0.971 - 0.236i)T \)
11 \( 1 + (-0.967 + 0.252i)T \)
13 \( 1 + (-0.687 - 0.726i)T \)
17 \( 1 + (0.933 + 0.358i)T \)
19 \( 1 + (0.467 + 0.884i)T \)
23 \( 1 + (-0.995 + 0.0955i)T \)
29 \( 1 + (0.921 - 0.388i)T \)
31 \( 1 + (0.336 + 0.941i)T \)
37 \( 1 + (-0.830 - 0.556i)T \)
41 \( 1 + (-0.260 - 0.965i)T \)
43 \( 1 + (0.908 + 0.417i)T \)
47 \( 1 + (0.998 - 0.0478i)T \)
53 \( 1 + (-0.763 + 0.645i)T \)
59 \( 1 + (-0.395 - 0.918i)T \)
61 \( 1 + (0.424 - 0.905i)T \)
67 \( 1 + (-0.963 - 0.267i)T \)
71 \( 1 + (0.721 + 0.692i)T \)
73 \( 1 + (-0.213 - 0.976i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.978 + 0.205i)T \)
89 \( 1 + (-0.522 - 0.852i)T \)
97 \( 1 + (-0.971 - 0.236i)T \)
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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

−18.498552241105928488838009456758, −18.01042909517089782164049038991, −17.0288496348364170550395563264, −16.44664764455581743366728088471, −15.802145466007456633073834271975, −15.35329487301416049067654619854, −14.51449399439374589226735967678, −13.92207529943471029314004602373, −13.46518871586404659850804749328, −12.43734690856269684771461699378, −11.54332800253380969355470232646, −10.573336284472055487832386418235, −9.99764751222032639101702125965, −9.74352372653155844687901224799, −9.048139399307690826775875651025, −8.35922032109201932841556326078, −7.42685442806058372139235932166, −7.04395312145142638783659216522, −6.11390116077061599712833986298, −5.4651416390583824655027033687, −4.64889001428666799635746877691, −3.3298494746709001741351040079, −2.59066757921307800106709892962, −2.397953043377321544423981444416, −1.1871352783694912730599290887, 0.32907729727974364170561920638, 1.28729159888638160574569398928, 2.02994916326459868157182330108, 2.8120472887673177308547559511, 3.27887962591405045730070027349, 4.254168814110256511552295199653, 5.54087887330165983288844922316, 6.14821622789753237913625104247, 7.0891629944342230852883986509, 7.707535703211373261092442784060, 8.25119188065198994679771878985, 8.98986155551394824800673175405, 9.8874069824646528756336698449, 9.99198633169236691264989853515, 10.57331619687501832383909577290, 12.19861542762485693283495537797, 12.46332770571872498264462703174, 12.803156284534404042395242030443, 13.87690020915907947160668022864, 14.159952530249598261999212349980, 15.51730323205620282085214490435, 15.8010084865068801265230052062, 16.60250838752575693231316252936, 17.43282711233819478203719786082, 17.81919711745686345892352888851

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