Properties

Label 2-2400-1.1-c1-0-3
Degree $2$
Conductor $2400$
Sign $1$
Analytic cond. $19.1640$
Root an. cond. $4.37768$
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 − 2·7-s + 9-s + 4.47·11-s − 4.47·13-s + 4.47·17-s + 2·21-s − 4·23-s − 27-s + 4·29-s − 8.94·31-s − 4.47·33-s + 4.47·37-s + 4.47·39-s + 10·41-s + 4·43-s − 8·47-s − 3·49-s − 4.47·51-s + 4.47·53-s − 13.4·59-s + 10·61-s − 2·63-s + 8·67-s + 4·69-s + 8.94·71-s + 8.94·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 0.333·9-s + 1.34·11-s − 1.24·13-s + 1.08·17-s + 0.436·21-s − 0.834·23-s − 0.192·27-s + 0.742·29-s − 1.60·31-s − 0.778·33-s + 0.735·37-s + 0.716·39-s + 1.56·41-s + 0.609·43-s − 1.16·47-s − 0.428·49-s − 0.626·51-s + 0.614·53-s − 1.74·59-s + 1.28·61-s − 0.251·63-s + 0.977·67-s + 0.481·69-s + 1.06·71-s + 1.04·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(19.1640\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.260776783\)
\(L(\frac12)\) \(\approx\) \(1.260776783\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7 \( 1 + 2T + 7T^{2} \)
11 \( 1 - 4.47T + 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 - 4.47T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + 8.94T + 31T^{2} \)
37 \( 1 - 4.47T + 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 4.47T + 53T^{2} \)
59 \( 1 + 13.4T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 - 8.94T + 71T^{2} \)
73 \( 1 - 8.94T + 73T^{2} \)
79 \( 1 - 8.94T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 17.8T + 97T^{2} \)
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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

−9.394923561499792875278883090299, −8.054586098693156195597943569590, −7.33687140914472755794897013319, −6.55678835939007687340089477271, −5.95339759999321565514956373542, −5.07083261314643721631974564941, −4.11514933551697702845382619408, −3.32972740334086786409784125902, −2.08109084610112438217265453635, −0.73889573525456845550463854024, 0.73889573525456845550463854024, 2.08109084610112438217265453635, 3.32972740334086786409784125902, 4.11514933551697702845382619408, 5.07083261314643721631974564941, 5.95339759999321565514956373542, 6.55678835939007687340089477271, 7.33687140914472755794897013319, 8.054586098693156195597943569590, 9.394923561499792875278883090299

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