Properties

Label 1-4729-4729.3485-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.965 - 0.261i$
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  + (−0.954 + 0.298i)2-s + (0.627 − 0.778i)3-s + (0.821 − 0.569i)4-s + (−0.285 + 0.958i)5-s + (−0.366 + 0.930i)6-s + (−0.0770 − 0.997i)7-s + (−0.614 + 0.788i)8-s + (−0.213 − 0.976i)9-s + (−0.0132 − 0.999i)10-s + (0.773 − 0.633i)11-s + (0.0717 − 0.997i)12-s + (0.777 + 0.629i)13-s + (0.370 + 0.928i)14-s + (0.567 + 0.823i)15-s + (0.351 − 0.936i)16-s + (−0.964 − 0.262i)17-s + ⋯
L(s)  = 1  + (−0.954 + 0.298i)2-s + (0.627 − 0.778i)3-s + (0.821 − 0.569i)4-s + (−0.285 + 0.958i)5-s + (−0.366 + 0.930i)6-s + (−0.0770 − 0.997i)7-s + (−0.614 + 0.788i)8-s + (−0.213 − 0.976i)9-s + (−0.0132 − 0.999i)10-s + (0.773 − 0.633i)11-s + (0.0717 − 0.997i)12-s + (0.777 + 0.629i)13-s + (0.370 + 0.928i)14-s + (0.567 + 0.823i)15-s + (0.351 − 0.936i)16-s + (−0.964 − 0.262i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.965 - 0.261i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (3485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.965 - 0.261i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05961153028 - 0.4474248257i\)
\(L(\frac12)\) \(\approx\) \(0.05961153028 - 0.4474248257i\)
\(L(1)\) \(\approx\) \(0.7164697792 - 0.1337693727i\)
\(L(1)\) \(\approx\) \(0.7164697792 - 0.1337693727i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.954 + 0.298i)T \)
3 \( 1 + (0.627 - 0.778i)T \)
5 \( 1 + (-0.285 + 0.958i)T \)
7 \( 1 + (-0.0770 - 0.997i)T \)
11 \( 1 + (0.773 - 0.633i)T \)
13 \( 1 + (0.777 + 0.629i)T \)
17 \( 1 + (-0.964 - 0.262i)T \)
19 \( 1 + (-0.0345 + 0.999i)T \)
23 \( 1 + (-0.135 + 0.990i)T \)
29 \( 1 + (0.954 + 0.298i)T \)
31 \( 1 + (0.908 - 0.417i)T \)
37 \( 1 + (-0.890 - 0.455i)T \)
41 \( 1 + (-0.721 - 0.692i)T \)
43 \( 1 + (-0.643 + 0.765i)T \)
47 \( 1 + (-0.946 - 0.323i)T \)
53 \( 1 + (0.545 + 0.838i)T \)
59 \( 1 + (-0.760 - 0.649i)T \)
61 \( 1 + (-0.683 - 0.730i)T \)
67 \( 1 + (-0.992 + 0.121i)T \)
71 \( 1 + (0.531 + 0.846i)T \)
73 \( 1 + (-0.818 + 0.573i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.527 + 0.849i)T \)
89 \( 1 + (-0.796 - 0.604i)T \)
97 \( 1 + (-0.0770 - 0.997i)T \)
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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

−18.465570988705877937099691428673, −17.74558743957810597848130508911, −17.1799505959650677113691085293, −16.38698430884965519836606060912, −15.79717571311852999673014239227, −15.32897133657856468959850873967, −14.90844691451655477151073259690, −13.57602339923120984327997484870, −13.06272275815213954472818361929, −12.09846693883919007397046776121, −11.78386213689525609353517398899, −10.83961888718846331862881344131, −10.177165267930792293866683413077, −9.44592515016201416216292273262, −8.72421567146059233180306011620, −8.623486582148286754538279084059, −7.96716625861728217293246995799, −6.793575015449931206686523803131, −6.16171386560224959825100518276, −4.96365321316906258396172859866, −4.47564534388877559997880449488, −3.52971787195986321607987461205, −2.78802279620749401676772146219, −2.02279403731264921489313560084, −1.21027557332215481408969559176, 0.150414415121223124997702178162, 1.35410361568505029870678707508, 1.753423235889802994531513787805, 2.93674531897702198768539754484, 3.49164831053547206365139771355, 4.273960996846041090620948602434, 5.85676034548760143453528038928, 6.52729618671665893800490811391, 6.82353207177547097765444848253, 7.52206689362262050447840497536, 8.24995413709379352767881233298, 8.750759738442831691234201098325, 9.63796943959083339240487094495, 10.25454712446528961478902984772, 11.13967680989706343305747378952, 11.51290506500313534440173188313, 12.25548038441383994482873336210, 13.5806640827674626321398304859, 13.90133221425658162389263093315, 14.3828604835661326912131325312, 15.26529508923980480297330726940, 15.842986020314479641374875104376, 16.63638164702680369263074812062, 17.37326036858576701231840477358, 17.9218937412024561942452113004

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