Properties

Label 2-2280-1.1-c1-0-9
Degree $2$
Conductor $2280$
Sign $1$
Analytic cond. $18.2058$
Root an. cond. $4.26683$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 1.29·7-s + 9-s − 5.01·11-s + 3.29·13-s − 15-s + 5.71·17-s − 19-s + 1.29·21-s − 3.71·23-s + 25-s + 27-s + 4.41·29-s + 7.71·31-s − 5.01·33-s − 1.29·35-s + 3.29·37-s + 3.29·39-s − 7.01·41-s + 5.29·43-s − 45-s + 3.71·47-s − 5.31·49-s + 5.71·51-s + 5.71·53-s + 5.01·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.491·7-s + 0.333·9-s − 1.51·11-s + 0.915·13-s − 0.258·15-s + 1.38·17-s − 0.229·19-s + 0.283·21-s − 0.773·23-s + 0.200·25-s + 0.192·27-s + 0.819·29-s + 1.38·31-s − 0.872·33-s − 0.219·35-s + 0.542·37-s + 0.528·39-s − 1.09·41-s + 0.808·43-s − 0.149·45-s + 0.541·47-s − 0.758·49-s + 0.799·51-s + 0.784·53-s + 0.675·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(18.2058\)
Root analytic conductor: \(4.26683\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.143355114\)
\(L(\frac12)\) \(\approx\) \(2.143355114\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 + T \)
19 \( 1 + T \)
good7 \( 1 - 1.29T + 7T^{2} \)
11 \( 1 + 5.01T + 11T^{2} \)
13 \( 1 - 3.29T + 13T^{2} \)
17 \( 1 - 5.71T + 17T^{2} \)
23 \( 1 + 3.71T + 23T^{2} \)
29 \( 1 - 4.41T + 29T^{2} \)
31 \( 1 - 7.71T + 31T^{2} \)
37 \( 1 - 3.29T + 37T^{2} \)
41 \( 1 + 7.01T + 41T^{2} \)
43 \( 1 - 5.29T + 43T^{2} \)
47 \( 1 - 3.71T + 47T^{2} \)
53 \( 1 - 5.71T + 53T^{2} \)
59 \( 1 - 6.59T + 59T^{2} \)
61 \( 1 - 7.11T + 61T^{2} \)
67 \( 1 - 11.4T + 67T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 - 16.5T + 83T^{2} \)
89 \( 1 + 9.61T + 89T^{2} \)
97 \( 1 - 11.2T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.744562064318574499243171830766, −8.009212005777289615762675737614, −7.940183865920907529661633920007, −6.81368976799422534631332608031, −5.79337705687277506358738772485, −5.00867922508602979870863409080, −4.08056402977940687716719507696, −3.18714946018868992964988309019, −2.31341029416785165323438360657, −0.952065895119842998882988728539, 0.952065895119842998882988728539, 2.31341029416785165323438360657, 3.18714946018868992964988309019, 4.08056402977940687716719507696, 5.00867922508602979870863409080, 5.79337705687277506358738772485, 6.81368976799422534631332608031, 7.940183865920907529661633920007, 8.009212005777289615762675737614, 8.744562064318574499243171830766

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