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Properties

Label 1-4725-4725.2981-r0-0-0
Degree $1$
Conductor $4725$
Sign $-0.806 - 0.591i$
Analytic cond. $21.9428$
Root an. cond. $21.9428$
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  + (0.719 + 0.694i)2-s + (0.0348 + 0.999i)4-s + (−0.669 + 0.743i)8-s + (0.241 − 0.970i)11-s + (0.719 − 0.694i)13-s + (−0.997 + 0.0697i)16-s + (−0.978 − 0.207i)17-s + (−0.669 + 0.743i)19-s + (0.848 − 0.529i)22-s + (−0.559 + 0.829i)23-s + 26-s + (−0.990 − 0.139i)29-s + (0.615 + 0.788i)31-s + (−0.766 − 0.642i)32-s + (−0.559 − 0.829i)34-s + ⋯
L(s)  = 1  + (0.719 + 0.694i)2-s + (0.0348 + 0.999i)4-s + (−0.669 + 0.743i)8-s + (0.241 − 0.970i)11-s + (0.719 − 0.694i)13-s + (−0.997 + 0.0697i)16-s + (−0.978 − 0.207i)17-s + (−0.669 + 0.743i)19-s + (0.848 − 0.529i)22-s + (−0.559 + 0.829i)23-s + 26-s + (−0.990 − 0.139i)29-s + (0.615 + 0.788i)31-s + (−0.766 − 0.642i)32-s + (−0.559 − 0.829i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4725\)    =    \(3^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.806 - 0.591i$
Analytic conductor: \(21.9428\)
Root analytic conductor: \(21.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4725} (2981, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4725,\ (0:\ ),\ -0.806 - 0.591i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1961096006 + 0.5988181208i\)
\(L(\frac12)\) \(\approx\) \(-0.1961096006 + 0.5988181208i\)
\(L(1)\) \(\approx\) \(1.074675666 + 0.5605946259i\)
\(L(1)\) \(\approx\) \(1.074675666 + 0.5605946259i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.719 + 0.694i)T \)
11 \( 1 + (0.241 - 0.970i)T \)
13 \( 1 + (0.719 - 0.694i)T \)
17 \( 1 + (-0.978 - 0.207i)T \)
19 \( 1 + (-0.669 + 0.743i)T \)
23 \( 1 + (-0.559 + 0.829i)T \)
29 \( 1 + (-0.990 - 0.139i)T \)
31 \( 1 + (0.615 + 0.788i)T \)
37 \( 1 + (0.913 - 0.406i)T \)
41 \( 1 + (-0.719 + 0.694i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (-0.615 + 0.788i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.241 - 0.970i)T \)
61 \( 1 + (-0.961 + 0.275i)T \)
67 \( 1 + (0.990 - 0.139i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (-0.913 - 0.406i)T \)
79 \( 1 + (0.990 + 0.139i)T \)
83 \( 1 + (-0.882 + 0.469i)T \)
89 \( 1 + (0.913 + 0.406i)T \)
97 \( 1 + (0.374 + 0.927i)T \)
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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

−17.90134071106127340807033337170, −17.16668807127953164281564666937, −16.28279351514670437709029479984, −15.53318578354699145573113205819, −14.9442836328531569777759718889, −14.438280991747536014327870190898, −13.45986976275379740992122591064, −13.15908130037904967928535104689, −12.39854796008714904625135538491, −11.63469804987257867869990922919, −11.14860085855853801608289897300, −10.43857540358669142823204173477, −9.67252532564448279266331147501, −9.06147039925787173730766951378, −8.32507623524471346053795532729, −7.13373001632156894398154310857, −6.525635468948811820053661472541, −5.987141544083803575647599762756, −4.910306288152710758115426572763, −4.28607240514442710248565726647, −3.907581255594365157815207924905, −2.709421775868559205242764992727, −2.097807448634706775255375783657, −1.42388433362001692182311623428, −0.11915305617483176940464632656, 1.32973772179062442756241638145, 2.40508624408359281137584378641, 3.28910699961869534431523468475, 3.82987842574820253602754030316, 4.60475966017102717836314794994, 5.51778605739523169557550379808, 6.10815568741502479057332151068, 6.55492243059758494607356622434, 7.648838298119651616701324376789, 8.1097527114968673735315640160, 8.82109661831890097843301882235, 9.50009427282034292382799254829, 10.73011644533967517680622826721, 11.14024418086737304067146250146, 11.955864753640281088672708119907, 12.67129882974802026898754254602, 13.4220944414322898964987045522, 13.74847395027999843850536708774, 14.57641583085565082571514426668, 15.23573803860280101823017845176, 15.88726737470210119293485311831, 16.35213845530052739141363692240, 17.14410953195208816638881087138, 17.72180273817631547601580175103, 18.409084499382482659380935824187

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