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Properties

Label 1-4729-4729.1048-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.786 - 0.617i$
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  + (−0.495 + 0.868i)2-s + (0.812 − 0.582i)3-s + (−0.509 − 0.860i)4-s + (0.859 − 0.511i)5-s + (0.103 + 0.994i)6-s + (0.671 + 0.741i)7-s + (0.999 − 0.0159i)8-s + (0.321 − 0.947i)9-s + (0.0186 + 0.999i)10-s + (−0.798 + 0.601i)11-s + (−0.915 − 0.402i)12-s + (−0.947 − 0.318i)13-s + (−0.976 + 0.216i)14-s + (0.400 − 0.916i)15-s + (−0.481 + 0.876i)16-s + (0.363 − 0.931i)17-s + ⋯
L(s)  = 1  + (−0.495 + 0.868i)2-s + (0.812 − 0.582i)3-s + (−0.509 − 0.860i)4-s + (0.859 − 0.511i)5-s + (0.103 + 0.994i)6-s + (0.671 + 0.741i)7-s + (0.999 − 0.0159i)8-s + (0.321 − 0.947i)9-s + (0.0186 + 0.999i)10-s + (−0.798 + 0.601i)11-s + (−0.915 − 0.402i)12-s + (−0.947 − 0.318i)13-s + (−0.976 + 0.216i)14-s + (0.400 − 0.916i)15-s + (−0.481 + 0.876i)16-s + (0.363 − 0.931i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.786 - 0.617i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1048, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.786 - 0.617i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.045872524 - 0.7076189218i\)
\(L(\frac12)\) \(\approx\) \(2.045872524 - 0.7076189218i\)
\(L(1)\) \(\approx\) \(1.272981373 + 0.005854697660i\)
\(L(1)\) \(\approx\) \(1.272981373 + 0.005854697660i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.495 + 0.868i)T \)
3 \( 1 + (0.812 - 0.582i)T \)
5 \( 1 + (0.859 - 0.511i)T \)
7 \( 1 + (0.671 + 0.741i)T \)
11 \( 1 + (-0.798 + 0.601i)T \)
13 \( 1 + (-0.947 - 0.318i)T \)
17 \( 1 + (0.363 - 0.931i)T \)
19 \( 1 + (0.964 - 0.262i)T \)
23 \( 1 + (-0.905 + 0.424i)T \)
29 \( 1 + (0.868 - 0.495i)T \)
31 \( 1 + (0.608 - 0.793i)T \)
37 \( 1 + (0.341 - 0.939i)T \)
41 \( 1 + (0.726 + 0.687i)T \)
43 \( 1 + (-0.616 + 0.787i)T \)
47 \( 1 + (-0.886 + 0.462i)T \)
53 \( 1 + (0.900 + 0.434i)T \)
59 \( 1 + (-0.935 - 0.353i)T \)
61 \( 1 + (0.673 + 0.739i)T \)
67 \( 1 + (0.897 + 0.441i)T \)
71 \( 1 + (-0.156 - 0.987i)T \)
73 \( 1 + (0.655 - 0.755i)T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (-0.986 + 0.164i)T \)
89 \( 1 + (-0.999 + 0.0345i)T \)
97 \( 1 + (-0.671 - 0.741i)T \)
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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

−18.36839458878782754796788686708, −17.6749724366574161953645297029, −16.96262201046213595931322276735, −16.458788529580594316680638102005, −15.58299267683478261121170707622, −14.55046147533481658416741720395, −14.06964570637838466224924020242, −13.73024934021651730915587001344, −12.92465506060635378580224205545, −12.086256412072008725905188450548, −11.16256937727202708601767807158, −10.540956317551783990891174633502, −10.00327477889246439522165479950, −9.79965472078256856124626180293, −8.59707873421559022378874906710, −8.22306201935446841575124416739, −7.50142245136404330902697421059, −6.738088944052784691232202414592, −5.360363089525223184580142161426, −4.86833984329560569483413659782, −3.92198581276089576027041909402, −3.261504347863943664331515466152, −2.54009185424877519513831527533, −1.913126248142590462781968353392, −1.086103194066074017396801589673, 0.63777932089886865286736771975, 1.52912433387203445528492350104, 2.32649365183351773760404170837, 2.808198659592992733043201054611, 4.44484259540266623955223858917, 4.98329863941628212517597709294, 5.671103562070043880817440527095, 6.333736841959023635642780326551, 7.39512322224142436804740339826, 7.76811183172396699388370313082, 8.332744516465603284682422026167, 9.21315929893535995811580180947, 9.75401832653550856626274332527, 10.013357425862699765285130085017, 11.40716891365256701704574409869, 12.21798902626163556108568340927, 12.871901773936044434575364763499, 13.64597781394437749712503201264, 14.1173902485218425692334782525, 14.72503928846734867583541721307, 15.42546973223361012384085556220, 15.954380702996132267505919441940, 16.83992399620908931739160441540, 17.77749172821494603531038317555, 18.02971876236431145210037971896

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