Properties

Label 2-2280-1.1-c1-0-0
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 − 2.77·7-s + 9-s − 0.778·11-s − 5.50·13-s + 15-s − 7.00·17-s + 19-s + 2.77·21-s − 1.27·23-s + 25-s − 27-s + 8.23·29-s + 5.00·31-s + 0.778·33-s + 2.77·35-s + 8.51·37-s + 5.50·39-s + 9.06·41-s − 10.5·43-s − 45-s − 6.72·47-s + 0.717·49-s + 7.00·51-s + 8.72·53-s + 0.778·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 1.04·7-s + 0.333·9-s − 0.234·11-s − 1.52·13-s + 0.258·15-s − 1.70·17-s + 0.229·19-s + 0.606·21-s − 0.265·23-s + 0.200·25-s − 0.192·27-s + 1.52·29-s + 0.899·31-s + 0.135·33-s + 0.469·35-s + 1.39·37-s + 0.881·39-s + 1.41·41-s − 1.60·43-s − 0.149·45-s − 0.981·47-s + 0.102·49-s + 0.981·51-s + 1.19·53-s + 0.104·55-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

\(L(1)\) \(\approx\) \(0.6394945861\)
\(L(\frac12)\) \(\approx\) \(0.6394945861\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.77T + 7T^{2} \)
11 \( 1 + 0.778T + 11T^{2} \)
13 \( 1 + 5.50T + 13T^{2} \)
17 \( 1 + 7.00T + 17T^{2} \)
23 \( 1 + 1.27T + 23T^{2} \)
29 \( 1 - 8.23T + 29T^{2} \)
31 \( 1 - 5.00T + 31T^{2} \)
37 \( 1 - 8.51T + 37T^{2} \)
41 \( 1 - 9.06T + 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 + 6.72T + 47T^{2} \)
53 \( 1 - 8.72T + 53T^{2} \)
59 \( 1 + 5.29T + 59T^{2} \)
61 \( 1 + 7.73T + 61T^{2} \)
67 \( 1 + 1.45T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 - 14.5T + 73T^{2} \)
79 \( 1 - 2.28T + 79T^{2} \)
83 \( 1 - 3.45T + 83T^{2} \)
89 \( 1 + 2.51T + 89T^{2} \)
97 \( 1 - 8.51T + 97T^{2} \)
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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

−9.143530725068651964054650001054, −8.179964310081354994672688224658, −7.34330198014524915459169362120, −6.61728795118162019768568763515, −6.09810116047250525231431237307, −4.79551937360616566535910273697, −4.46336391010847641818491404612, −3.13419697385842829772938121604, −2.32292200278481330769654759971, −0.50219645841858026346765490631, 0.50219645841858026346765490631, 2.32292200278481330769654759971, 3.13419697385842829772938121604, 4.46336391010847641818491404612, 4.79551937360616566535910273697, 6.09810116047250525231431237307, 6.61728795118162019768568763515, 7.34330198014524915459169362120, 8.179964310081354994672688224658, 9.143530725068651964054650001054

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