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Properties

Label 1-4729-4729.105-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.980 + 0.195i$
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  + (−0.380 − 0.924i)2-s + (−0.987 + 0.158i)3-s + (−0.709 + 0.704i)4-s + (0.655 + 0.755i)5-s + (0.522 + 0.852i)6-s + (−0.429 − 0.903i)7-s + (0.921 + 0.388i)8-s + (0.949 − 0.313i)9-s + (0.448 − 0.893i)10-s + (0.905 + 0.424i)11-s + (0.589 − 0.808i)12-s + (0.249 + 0.968i)13-s + (−0.671 + 0.741i)14-s + (−0.767 − 0.641i)15-s + (0.00797 − 0.999i)16-s + (0.121 − 0.992i)17-s + ⋯
L(s)  = 1  + (−0.380 − 0.924i)2-s + (−0.987 + 0.158i)3-s + (−0.709 + 0.704i)4-s + (0.655 + 0.755i)5-s + (0.522 + 0.852i)6-s + (−0.429 − 0.903i)7-s + (0.921 + 0.388i)8-s + (0.949 − 0.313i)9-s + (0.448 − 0.893i)10-s + (0.905 + 0.424i)11-s + (0.589 − 0.808i)12-s + (0.249 + 0.968i)13-s + (−0.671 + 0.741i)14-s + (−0.767 − 0.641i)15-s + (0.00797 − 0.999i)16-s + (0.121 − 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.980 + 0.195i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.980 + 0.195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.220228317 + 0.1203236287i\)
\(L(\frac12)\) \(\approx\) \(1.220228317 + 0.1203236287i\)
\(L(1)\) \(\approx\) \(0.7827753729 - 0.1229845451i\)
\(L(1)\) \(\approx\) \(0.7827753729 - 0.1229845451i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.380 - 0.924i)T \)
3 \( 1 + (-0.987 + 0.158i)T \)
5 \( 1 + (0.655 + 0.755i)T \)
7 \( 1 + (-0.429 - 0.903i)T \)
11 \( 1 + (0.905 + 0.424i)T \)
13 \( 1 + (0.249 + 0.968i)T \)
17 \( 1 + (0.121 - 0.992i)T \)
19 \( 1 + (0.935 + 0.353i)T \)
23 \( 1 + (0.0318 + 0.999i)T \)
29 \( 1 + (0.924 + 0.380i)T \)
31 \( 1 + (-0.595 - 0.803i)T \)
37 \( 1 + (0.659 - 0.751i)T \)
41 \( 1 + (0.996 - 0.0875i)T \)
43 \( 1 + (0.785 + 0.618i)T \)
47 \( 1 + (-0.857 + 0.513i)T \)
53 \( 1 + (0.231 + 0.972i)T \)
59 \( 1 + (0.925 + 0.378i)T \)
61 \( 1 + (-0.368 - 0.929i)T \)
67 \( 1 + (0.419 + 0.907i)T \)
71 \( 1 + (0.702 - 0.711i)T \)
73 \( 1 + (-0.827 + 0.560i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (0.558 + 0.829i)T \)
89 \( 1 + (-0.649 - 0.760i)T \)
97 \( 1 + (0.429 + 0.903i)T \)
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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

−17.96288750445343769777765179621, −17.461920692574184435068581183515, −16.763867218159131567698712158816, −16.24109354788412709189136052530, −15.77681878053410075129463516542, −14.964603976389061606891901262086, −14.20196081933107219969876268195, −13.32683530088384164029346722090, −12.80353088602840682336592012269, −12.24549583602084180746819308189, −11.38306435995788057093965133455, −10.41061433038288579389271003947, −9.94889652462652571633284669863, −9.14923396676945064051194638981, −8.571251554380821879307569381450, −7.92358473507521312064786494709, −6.80787056916612111145019957452, −6.228045624161248629995418083246, −5.81815138211150044595820505662, −5.194042059605883549759456477036, −4.536821975010243300464166027875, −3.49689644643996527453153739874, −2.161233178858921339093087856561, −1.17486142791072325759258612923, −0.62557440757102725087727086433, 0.927449654404807213593935318475, 1.42448173480580630315877811711, 2.43393476880913021929668117845, 3.44104500110437770478318228187, 4.00943063134068999338204577957, 4.72034196372111971314607598790, 5.68892842965273756181787907679, 6.48265056428325577883198042992, 7.23060690290491580858677744998, 7.5688817052973100872138668733, 9.21609396568717349410026925379, 9.58486234497595324692295098753, 9.91519698546846207023045603652, 10.93867445648342469838629601742, 11.2277640111881529917032913133, 11.87646109701533570161827315039, 12.633368690938299448398775143707, 13.41542513211924563289965984989, 14.03612283208649267968726936114, 14.501833121598151089122014204883, 15.8674300214470984300917997773, 16.43688538754398340451586301863, 16.97239657236731271661763951594, 17.76848152796346357895405059348, 17.972855644704152443822342837881

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