Properties

Label 2-1200-1.1-c3-0-21
Degree $2$
Conductor $1200$
Sign $1$
Analytic cond. $70.8022$
Root an. cond. $8.41440$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 33.0·7-s + 9·9-s − 48.3·11-s − 60.3·13-s − 17.7·17-s + 130.·19-s + 99.2·21-s − 70.8·23-s + 27·27-s + 104.·29-s + 210.·31-s − 145.·33-s + 300.·37-s − 181.·39-s + 240.·41-s + 108·43-s + 278.·47-s + 750.·49-s − 53.3·51-s + 328.·53-s + 392.·57-s − 889.·59-s − 241.·61-s + 297.·63-s − 103.·67-s − 212.·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.78·7-s + 0.333·9-s − 1.32·11-s − 1.28·13-s − 0.253·17-s + 1.58·19-s + 1.03·21-s − 0.642·23-s + 0.192·27-s + 0.669·29-s + 1.21·31-s − 0.765·33-s + 1.33·37-s − 0.743·39-s + 0.914·41-s + 0.383·43-s + 0.865·47-s + 2.18·49-s − 0.146·51-s + 0.851·53-s + 0.912·57-s − 1.96·59-s − 0.506·61-s + 0.595·63-s − 0.189·67-s − 0.370·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(70.8022\)
Root analytic conductor: \(8.41440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.119990509\)
\(L(\frac12)\) \(\approx\) \(3.119990509\)
\(L(\frac{5}{2})\) 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 \( 1 - 3T \)
5 \( 1 \)
good7 \( 1 - 33.0T + 343T^{2} \)
11 \( 1 + 48.3T + 1.33e3T^{2} \)
13 \( 1 + 60.3T + 2.19e3T^{2} \)
17 \( 1 + 17.7T + 4.91e3T^{2} \)
19 \( 1 - 130.T + 6.85e3T^{2} \)
23 \( 1 + 70.8T + 1.21e4T^{2} \)
29 \( 1 - 104.T + 2.43e4T^{2} \)
31 \( 1 - 210.T + 2.97e4T^{2} \)
37 \( 1 - 300.T + 5.06e4T^{2} \)
41 \( 1 - 240.T + 6.89e4T^{2} \)
43 \( 1 - 108T + 7.95e4T^{2} \)
47 \( 1 - 278.T + 1.03e5T^{2} \)
53 \( 1 - 328.T + 1.48e5T^{2} \)
59 \( 1 + 889.T + 2.05e5T^{2} \)
61 \( 1 + 241.T + 2.26e5T^{2} \)
67 \( 1 + 103.T + 3.00e5T^{2} \)
71 \( 1 - 277.T + 3.57e5T^{2} \)
73 \( 1 - 274.T + 3.89e5T^{2} \)
79 \( 1 + 366.T + 4.93e5T^{2} \)
83 \( 1 + 57.7T + 5.71e5T^{2} \)
89 \( 1 + 203.T + 7.04e5T^{2} \)
97 \( 1 - 1.28e3T + 9.12e5T^{2} \)
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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

−9.372783420025994038417886062849, −8.340661919045081631091633034023, −7.68880235023874252580015271091, −7.42556424170616120194743333112, −5.82023630005094069514386690949, −4.90238083442971768066309319949, −4.44863420620650766132696366503, −2.84867373886782349622594371706, −2.19081336113120710667779492294, −0.902478242542447817355648283085, 0.902478242542447817355648283085, 2.19081336113120710667779492294, 2.84867373886782349622594371706, 4.44863420620650766132696366503, 4.90238083442971768066309319949, 5.82023630005094069514386690949, 7.42556424170616120194743333112, 7.68880235023874252580015271091, 8.340661919045081631091633034023, 9.372783420025994038417886062849

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