Properties

Label 1-2280-2280.1229-r0-0-0
Degree $1$
Conductor $2280$
Sign $-0.999 + 0.000652i$
Analytic cond. $10.5882$
Root an. cond. $10.5882$
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  + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)11-s + (−0.766 + 0.642i)13-s + (0.173 − 0.984i)17-s + (−0.939 − 0.342i)23-s + (−0.173 − 0.984i)29-s + (0.5 + 0.866i)31-s − 37-s + (0.766 + 0.642i)41-s + (−0.939 + 0.342i)43-s + (0.173 + 0.984i)47-s + (−0.5 + 0.866i)49-s + (−0.939 − 0.342i)53-s + (−0.173 + 0.984i)59-s + (0.939 + 0.342i)61-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)11-s + (−0.766 + 0.642i)13-s + (0.173 − 0.984i)17-s + (−0.939 − 0.342i)23-s + (−0.173 − 0.984i)29-s + (0.5 + 0.866i)31-s − 37-s + (0.766 + 0.642i)41-s + (−0.939 + 0.342i)43-s + (0.173 + 0.984i)47-s + (−0.5 + 0.866i)49-s + (−0.939 − 0.342i)53-s + (−0.173 + 0.984i)59-s + (0.939 + 0.342i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.000652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.000652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.999 + 0.000652i$
Analytic conductor: \(10.5882\)
Root analytic conductor: \(10.5882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (1229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2280,\ (0:\ ),\ -0.999 + 0.000652i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0001462463577 + 0.4481250490i\)
\(L(\frac12)\) \(\approx\) \(0.0001462463577 + 0.4481250490i\)
\(L(1)\) \(\approx\) \(0.8364673002 + 0.1940814634i\)
\(L(1)\) \(\approx\) \(0.8364673002 + 0.1940814634i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (-0.173 - 0.984i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (-0.173 + 0.984i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (-0.766 - 0.642i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−19.25458464317587583927607348015, −18.696920004516854566086345066518, −17.59146401044600709394429296711, −17.34258472334096859448900147919, −16.45113936692234308482245266342, −15.765185537088641395104265089649, −14.87100887403369171181352879158, −14.26201048402898819728852339039, −13.513523441819791274002619908289, −12.862963118976358999193307398230, −11.99992856767814023352935291044, −11.17920881384614870916929191724, −10.4074807454220098083066587567, −10.04719746661277656822190431261, −8.824403020342323889552880577423, −8.06172844563664415628051591155, −7.57451884497143610609408511011, −6.63118605264704025277604868737, −5.66017209166230780735359256, −5.0533004127885411969777073676, −3.99672840938045513295647444909, −3.37030392132688862434736254882, −2.27323173580326298910178575347, −1.30505737570784822655482158854, −0.14096170013169358009200394205, 1.54347320868836884191678642470, 2.32125867857619535069717683891, 2.99425198177722605530789148877, 4.4191932446070159860785569143, 4.82865823042247655806750107921, 5.7057847502727486265019838306, 6.6201366842663039954674718918, 7.48674069068610566845413409931, 8.10403712613567856048979738020, 9.05821888132947919307825886661, 9.70620412523435493252001746806, 10.39084329843544809619599498904, 11.512385788139567469852817246082, 11.98457674463448008403890337832, 12.57867199762948645244072813073, 13.57712369274960907760394787743, 14.377548138260478124387605381752, 14.89287602380199890226880196842, 15.77388201588092355990285703781, 16.24147849670615373011811275737, 17.37667113542381296076175691463, 17.83167159042594672920599165310, 18.55424028906619416624264966319, 19.21814405083734468632939794392, 20.03396590740158559996294457384

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