↑

Properties

Label 1-4760-4760.1363-r0-0-0
Degree $1$
Conductor $4760$
Sign $0.999 + 0.0295i$
Analytic cond. $22.1053$
Root an. cond. $22.1053$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.608 + 0.793i)3-s + (−0.258 + 0.965i)9-s + (0.130 − 0.991i)11-s − 13-s + (−0.965 − 0.258i)19-s + (−0.608 + 0.793i)23-s + (−0.923 + 0.382i)27-s + (0.382 − 0.923i)29-s + (0.793 − 0.608i)31-s + (0.866 − 0.5i)33-s + (0.991 − 0.130i)37-s + (−0.608 − 0.793i)39-s + (0.382 + 0.923i)41-s + (−0.707 + 0.707i)43-s + (−0.5 − 0.866i)47-s + ⋯
L(s)  = 1  + (0.608 + 0.793i)3-s + (−0.258 + 0.965i)9-s + (0.130 − 0.991i)11-s − 13-s + (−0.965 − 0.258i)19-s + (−0.608 + 0.793i)23-s + (−0.923 + 0.382i)27-s + (0.382 − 0.923i)29-s + (0.793 − 0.608i)31-s + (0.866 − 0.5i)33-s + (0.991 − 0.130i)37-s + (−0.608 − 0.793i)39-s + (0.382 + 0.923i)41-s + (−0.707 + 0.707i)43-s + (−0.5 − 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4760\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 17\)
Sign: $0.999 + 0.0295i$
Analytic conductor: \(22.1053\)
Root analytic conductor: \(22.1053\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4760} (1363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4760,\ (0:\ ),\ 0.999 + 0.0295i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.755546318 + 0.02590609475i\)
\(L(\frac12)\) \(\approx\) \(1.755546318 + 0.02590609475i\)
\(L(1)\) \(\approx\) \(1.158409411 + 0.2033678125i\)
\(L(1)\) \(\approx\) \(1.158409411 + 0.2033678125i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
17 \( 1 \)
good3 \( 1 + (0.608 + 0.793i)T \)
11 \( 1 + (0.130 - 0.991i)T \)
13 \( 1 - T \)
19 \( 1 + (-0.965 - 0.258i)T \)
23 \( 1 + (-0.608 + 0.793i)T \)
29 \( 1 + (0.382 - 0.923i)T \)
31 \( 1 + (0.793 - 0.608i)T \)
37 \( 1 + (0.991 - 0.130i)T \)
41 \( 1 + (0.382 + 0.923i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.965 - 0.258i)T \)
59 \( 1 + (0.965 - 0.258i)T \)
61 \( 1 + (0.608 - 0.793i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (-0.923 - 0.382i)T \)
73 \( 1 + (-0.793 + 0.608i)T \)
79 \( 1 + (0.793 + 0.608i)T \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.923 + 0.382i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.06822843137227608509305865159, −17.63976138861893586797971217023, −16.97148030283628201447920183032, −16.14389157542078930716644724028, −15.241912449169152286978116474050, −14.52951389466027305019138754726, −14.43006336108186577568706286191, −13.3072199232533909753942797035, −12.799393344688583842554885846405, −12.105922956429721048014952407061, −11.78684000197812842299726159576, −10.408492266583276234149395551053, −10.101209536902012573588604010323, −9.086157831497573707085527597453, −8.594622412994965940868055896429, −7.76637401723248087913814693656, −7.15686732546222541191694785424, −6.5942509235790305382776821145, −5.813033055498045292961604192897, −4.71857708477423764871776075464, −4.17994619468583927241567159694, −3.126403867055457984659795008716, −2.34890863240338182910146132141, −1.861285791567378706457755490120, −0.789696332389592995525753948305, 0.528855390883200063291206195, 1.92319779748093850051725771467, 2.60914801610248087387108456097, 3.3150353332995510832573129407, 4.17997578073553460411934788144, 4.686288038417977093777398221106, 5.6154563817295776377381429012, 6.27453463064223135518176428597, 7.259241463191902310600337932064, 8.25321807898247159221739419125, 8.32844717790817606030440466192, 9.55995341648080574682007218169, 9.7562118978850166335379228975, 10.61154324559300173469947720028, 11.39628746040788172645625076370, 11.858807285847322936297029109694, 13.05247535778349040572859916437, 13.476842474103208008789921796935, 14.23312273568005257069864479572, 14.904927049217681267574108979089, 15.315454992962813686087346055017, 16.2175875672272842103155315526, 16.670120583691072225808849829023, 17.32298816890857693795951814985, 18.13716165597239889885357485410

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