Properties

Degree $2$
Conductor $1200$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5·7-s + 9-s + 6·11-s − 3·13-s − 2·17-s − 19-s − 5·21-s + 2·23-s − 27-s + 6·29-s − 3·31-s − 6·33-s − 6·37-s + 3·39-s + 4·41-s − 11·43-s + 10·47-s + 18·49-s + 2·51-s − 8·53-s + 57-s + 6·59-s + 3·61-s + 5·63-s + 67-s − 2·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.88·7-s + 1/3·9-s + 1.80·11-s − 0.832·13-s − 0.485·17-s − 0.229·19-s − 1.09·21-s + 0.417·23-s − 0.192·27-s + 1.11·29-s − 0.538·31-s − 1.04·33-s − 0.986·37-s + 0.480·39-s + 0.624·41-s − 1.67·43-s + 1.45·47-s + 18/7·49-s + 0.280·51-s − 1.09·53-s + 0.132·57-s + 0.781·59-s + 0.384·61-s + 0.629·63-s + 0.122·67-s − 0.240·69-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: $1$
Motivic weight: \(1\)
Character: $\chi_{1200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.808587506\)
\(L(\frac12)\) \(\approx\) \(1.808587506\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−9.742556924472679851874567668648, −8.862335264795745226913956586656, −8.181591334619364053526366201129, −7.15055493829120797623951709422, −6.53469172483539810785185446287, −5.31489759350842036417474055261, −4.69239385827948134429691285836, −3.88602063887419709715536750668, −2.12092304549440650764688814032, −1.15555668548223589947262730285, 1.15555668548223589947262730285, 2.12092304549440650764688814032, 3.88602063887419709715536750668, 4.69239385827948134429691285836, 5.31489759350842036417474055261, 6.53469172483539810785185446287, 7.15055493829120797623951709422, 8.181591334619364053526366201129, 8.862335264795745226913956586656, 9.742556924472679851874567668648

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