Properties

Label 2-1200-1.1-c1-0-9
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 + 3·13-s − 6·17-s + 7·19-s + 3·21-s + 6·23-s + 27-s − 2·29-s + 5·31-s − 2·33-s − 10·37-s + 3·39-s + 12·41-s + 3·43-s − 10·47-s + 2·49-s − 6·51-s + 7·57-s + 6·59-s − 13·61-s + 3·63-s + 7·67-s + 6·69-s + 4·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.13·7-s + 1/3·9-s − 0.603·11-s + 0.832·13-s − 1.45·17-s + 1.60·19-s + 0.654·21-s + 1.25·23-s + 0.192·27-s − 0.371·29-s + 0.898·31-s − 0.348·33-s − 1.64·37-s + 0.480·39-s + 1.87·41-s + 0.457·43-s − 1.45·47-s + 2/7·49-s − 0.840·51-s + 0.927·57-s + 0.781·59-s − 1.66·61-s + 0.377·63-s + 0.855·67-s + 0.722·69-s + 0.474·71-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.339596355\)
\(L(\frac12)\) \(\approx\) \(2.339596355\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
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

−9.550056288109099143931230152889, −8.878767058679084531028586330884, −8.137101642137297659611805626833, −7.46821861039588007675945788520, −6.54444166819099512610966331729, −5.29967500393029178357270106346, −4.65945797122965052180088814014, −3.50874630517236808113026847602, −2.43634464161106960470616484114, −1.25044768868847065733824862174, 1.25044768868847065733824862174, 2.43634464161106960470616484114, 3.50874630517236808113026847602, 4.65945797122965052180088814014, 5.29967500393029178357270106346, 6.54444166819099512610966331729, 7.46821861039588007675945788520, 8.137101642137297659611805626833, 8.878767058679084531028586330884, 9.550056288109099143931230152889

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