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Properties

Label 1-4725-4725.2246-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: $\chi_{4725} (2246, \cdot )$
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

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

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