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Properties

Label 1-4729-4729.30-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.611 + 0.791i$
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 + (−0.361 − 0.932i)5-s + (0.839 + 0.543i)6-s + (−0.931 − 0.363i)7-s + (0.687 + 0.726i)8-s + (0.821 + 0.569i)9-s + (−0.0981 − 0.995i)10-s + (−0.0478 − 0.998i)11-s + (0.663 + 0.748i)12-s + (0.300 + 0.953i)13-s + (−0.800 − 0.599i)14-s + (−0.0663 − 0.997i)15-s + (0.467 + 0.884i)16-s + (0.922 + 0.385i)17-s + ⋯
L(s)  = 1  + (0.963 + 0.267i)2-s + (0.954 + 0.298i)3-s + (0.856 + 0.516i)4-s + (−0.361 − 0.932i)5-s + (0.839 + 0.543i)6-s + (−0.931 − 0.363i)7-s + (0.687 + 0.726i)8-s + (0.821 + 0.569i)9-s + (−0.0981 − 0.995i)10-s + (−0.0478 − 0.998i)11-s + (0.663 + 0.748i)12-s + (0.300 + 0.953i)13-s + (−0.800 − 0.599i)14-s + (−0.0663 − 0.997i)15-s + (0.467 + 0.884i)16-s + (0.922 + 0.385i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.611 + 0.791i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.611 + 0.791i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.984012587 + 1.957349738i\)
\(L(\frac12)\) \(\approx\) \(3.984012587 + 1.957349738i\)
\(L(1)\) \(\approx\) \(2.315040579 + 0.5189568393i\)
\(L(1)\) \(\approx\) \(2.315040579 + 0.5189568393i\)

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 \( 1 + (-0.361 - 0.932i)T \)
7 \( 1 + (-0.931 - 0.363i)T \)
11 \( 1 + (-0.0478 - 0.998i)T \)
13 \( 1 + (0.300 + 0.953i)T \)
17 \( 1 + (0.922 + 0.385i)T \)
19 \( 1 + (-0.998 + 0.0584i)T \)
23 \( 1 + (0.930 + 0.366i)T \)
29 \( 1 + (0.267 + 0.963i)T \)
31 \( 1 + (-0.936 + 0.351i)T \)
37 \( 1 + (-0.0451 + 0.998i)T \)
41 \( 1 + (0.595 - 0.803i)T \)
43 \( 1 + (-0.989 - 0.145i)T \)
47 \( 1 + (-0.0743 - 0.997i)T \)
53 \( 1 + (0.991 + 0.129i)T \)
59 \( 1 + (0.108 + 0.994i)T \)
61 \( 1 + (0.857 - 0.513i)T \)
67 \( 1 + (0.962 - 0.272i)T \)
71 \( 1 + (-0.370 - 0.928i)T \)
73 \( 1 + (-0.187 + 0.982i)T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (-0.999 - 0.0265i)T \)
89 \( 1 + (0.755 + 0.655i)T \)
97 \( 1 + (0.931 + 0.363i)T \)
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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

−18.48632759000894230426964283273, −17.52468691609836003061121624514, −16.36708907363743265891881481194, −15.67947525994835632761412660644, −15.12945190636959423679501513351, −14.73508080142128507098428690414, −14.18388884825580357641402217964, −13.05695778972747614425989574622, −12.96055588892352228392390715073, −12.260576180171737537714045018079, −11.44631277517201076128927689433, −10.54900034344327475456909202289, −9.98825636092685755939190144399, −9.43758095400989032621796626155, −8.29442199927973468319043684490, −7.51426867959747006029682610283, −7.00205516774181849376571925098, −6.349355712698108443411212927662, −5.62388317094042014590036847084, −4.52124225112670696207198597257, −3.75641451821237123993323112279, −3.19825665946639457050484554306, −2.56859443511083699504398784253, −2.06109627006204818711816567276, −0.753315992831833549816691701048, 1.10704880608621896721387472464, 1.948995351935858292469550996600, 3.02944928730166346100969889880, 3.66811891431975819209417802292, 3.953511052238903617712884477885, 4.93756019994537819850776730217, 5.56521026807957279784553967113, 6.57171191750221475831684132351, 7.13471493978327881605790694522, 7.99686062878624836872482745894, 8.70957006123810441955914453787, 9.06963811536128741879011153141, 10.18571410136666924706968368613, 10.80978182624523251564061074391, 11.71451938070053581599593269010, 12.48203021894207810118942288557, 13.10734528864441633975222299586, 13.46805582484811418586869177535, 14.19742247965976416039846821848, 14.84597308506361883129164147050, 15.587599816625781436386269091504, 16.18748778302723667168887841049, 16.70972863522221964026261108812, 16.93981305056548462424660845722, 18.59212357472936345026441360054

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