Properties

Label 2-1200-1.1-c1-0-8
Degree $2$
Conductor $1200$
Sign $1$
Analytic cond. $9.58204$
Root an. cond. $3.09548$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3·7-s + 9-s − 2·11-s + 13-s + 2·17-s + 5·19-s + 3·21-s − 6·23-s + 27-s + 10·29-s + 3·31-s − 2·33-s + 2·37-s + 39-s − 8·41-s − 43-s − 2·47-s + 2·49-s + 2·51-s − 4·53-s + 5·57-s + 10·59-s + 7·61-s + 3·63-s + 3·67-s − 6·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.13·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 0.485·17-s + 1.14·19-s + 0.654·21-s − 1.25·23-s + 0.192·27-s + 1.85·29-s + 0.538·31-s − 0.348·33-s + 0.328·37-s + 0.160·39-s − 1.24·41-s − 0.152·43-s − 0.291·47-s + 2/7·49-s + 0.280·51-s − 0.549·53-s + 0.662·57-s + 1.30·59-s + 0.896·61-s + 0.377·63-s + 0.366·67-s − 0.722·69-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.334766374\)
\(L(\frac12)\) \(\approx\) \(2.334766374\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−9.946237306464798320060676890295, −8.686228257120952969207606021954, −8.134034752267247325522916900531, −7.57404393602003080106109524301, −6.47388999507393793454845833318, −5.34490029281998815763412767656, −4.63718444572594903228967418102, −3.50818980571584578672789924466, −2.44041792194616260799140408514, −1.24790523142424415923064013417, 1.24790523142424415923064013417, 2.44041792194616260799140408514, 3.50818980571584578672789924466, 4.63718444572594903228967418102, 5.34490029281998815763412767656, 6.47388999507393793454845833318, 7.57404393602003080106109524301, 8.134034752267247325522916900531, 8.686228257120952969207606021954, 9.946237306464798320060676890295

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