Properties

Label 2-1200-1.1-c3-0-56
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 22.2·7-s + 9·9-s + 1.79·11-s − 58.2·13-s − 18.9·17-s − 104.·19-s + 66.6·21-s − 49.6·23-s + 27·27-s − 293.·29-s − 64.4·31-s + 5.37·33-s + 19.8·37-s − 174.·39-s − 165.·41-s − 247.·43-s + 384.·47-s + 150.·49-s − 56.9·51-s − 463.·53-s − 314.·57-s + 73.7·59-s − 137.·61-s + 199.·63-s + 173.·67-s − 148.·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.19·7-s + 0.333·9-s + 0.0490·11-s − 1.24·13-s − 0.270·17-s − 1.26·19-s + 0.692·21-s − 0.449·23-s + 0.192·27-s − 1.87·29-s − 0.373·31-s + 0.0283·33-s + 0.0883·37-s − 0.716·39-s − 0.630·41-s − 0.877·43-s + 1.19·47-s + 0.438·49-s − 0.156·51-s − 1.20·53-s − 0.730·57-s + 0.162·59-s − 0.288·61-s + 0.399·63-s + 0.317·67-s − 0.259·69-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: $\chi_{1200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1200,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 22.2T + 343T^{2} \)
11 \( 1 - 1.79T + 1.33e3T^{2} \)
13 \( 1 + 58.2T + 2.19e3T^{2} \)
17 \( 1 + 18.9T + 4.91e3T^{2} \)
19 \( 1 + 104.T + 6.85e3T^{2} \)
23 \( 1 + 49.6T + 1.21e4T^{2} \)
29 \( 1 + 293.T + 2.43e4T^{2} \)
31 \( 1 + 64.4T + 2.97e4T^{2} \)
37 \( 1 - 19.8T + 5.06e4T^{2} \)
41 \( 1 + 165.T + 6.89e4T^{2} \)
43 \( 1 + 247.T + 7.95e4T^{2} \)
47 \( 1 - 384.T + 1.03e5T^{2} \)
53 \( 1 + 463.T + 1.48e5T^{2} \)
59 \( 1 - 73.7T + 2.05e5T^{2} \)
61 \( 1 + 137.T + 2.26e5T^{2} \)
67 \( 1 - 173.T + 3.00e5T^{2} \)
71 \( 1 - 594.T + 3.57e5T^{2} \)
73 \( 1 + 320.T + 3.89e5T^{2} \)
79 \( 1 - 770.T + 4.93e5T^{2} \)
83 \( 1 - 173.T + 5.71e5T^{2} \)
89 \( 1 - 1.01e3T + 7.04e5T^{2} \)
97 \( 1 + 384.T + 9.12e5T^{2} \)
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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

−8.932228356681616004460172006328, −8.073413254060382002252391792765, −7.54485599088918513206111052289, −6.60614180145211059896373100983, −5.38574824784766810406018975284, −4.61724493613106626815371744599, −3.74636879365726534589709906161, −2.35605267086697107036003493447, −1.72066153632002513608749342338, 0, 1.72066153632002513608749342338, 2.35605267086697107036003493447, 3.74636879365726534589709906161, 4.61724493613106626815371744599, 5.38574824784766810406018975284, 6.60614180145211059896373100983, 7.54485599088918513206111052289, 8.073413254060382002252391792765, 8.932228356681616004460172006328

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