Properties

Label 2-1200-3.2-c2-0-16
Degree $2$
Conductor $1200$
Sign $1$
Analytic cond. $32.6976$
Root an. cond. $5.71818$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 13·7-s + 9·9-s − 23·13-s − 11·19-s + 39·21-s − 27·27-s − 59·31-s − 26·37-s + 69·39-s + 83·43-s + 120·49-s + 33·57-s − 121·61-s − 117·63-s − 13·67-s + 46·73-s + 142·79-s + 81·81-s + 299·91-s + 177·93-s − 167·97-s + 194·103-s + 71·109-s + 78·111-s − 207·117-s + ⋯
L(s)  = 1  − 3-s − 1.85·7-s + 9-s − 1.76·13-s − 0.578·19-s + 13/7·21-s − 27-s − 1.90·31-s − 0.702·37-s + 1.76·39-s + 1.93·43-s + 2.44·49-s + 0.578·57-s − 1.98·61-s − 1.85·63-s − 0.194·67-s + 0.630·73-s + 1.79·79-s + 81-s + 23/7·91-s + 1.90·93-s − 1.72·97-s + 1.88·103-s + 0.651·109-s + 0.702·111-s − 1.76·117-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1200} (401, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3935822416\)
\(L(\frac12)\) \(\approx\) \(0.3935822416\)
\(L(2)\) 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 \( 1 + p T \)
5 \( 1 \)
good7 \( 1 + 13 T + p^{2} T^{2} \)
11 \( ( 1 - p T )( 1 + p T ) \)
13 \( 1 + 23 T + p^{2} T^{2} \)
17 \( ( 1 - p T )( 1 + p T ) \)
19 \( 1 + 11 T + p^{2} T^{2} \)
23 \( ( 1 - p T )( 1 + p T ) \)
29 \( ( 1 - p T )( 1 + p T ) \)
31 \( 1 + 59 T + p^{2} T^{2} \)
37 \( 1 + 26 T + p^{2} T^{2} \)
41 \( ( 1 - p T )( 1 + p T ) \)
43 \( 1 - 83 T + p^{2} T^{2} \)
47 \( ( 1 - p T )( 1 + p T ) \)
53 \( ( 1 - p T )( 1 + p T ) \)
59 \( ( 1 - p T )( 1 + p T ) \)
61 \( 1 + 121 T + p^{2} T^{2} \)
67 \( 1 + 13 T + p^{2} T^{2} \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( 1 - 46 T + p^{2} T^{2} \)
79 \( 1 - 142 T + p^{2} T^{2} \)
83 \( ( 1 - p T )( 1 + p T ) \)
89 \( ( 1 - p T )( 1 + p T ) \)
97 \( 1 + 167 T + p^{2} T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.577717426479420368779515890266, −9.158376545900250033237811636814, −7.51088355831290791095839984776, −7.02704569181704601844562747970, −6.19489569220104366113582384470, −5.47178115452758990275970262476, −4.44406817734147346073763160895, −3.41241412027901598510585152748, −2.21912643709397653505131714573, −0.36840013993712410346406489089, 0.36840013993712410346406489089, 2.21912643709397653505131714573, 3.41241412027901598510585152748, 4.44406817734147346073763160895, 5.47178115452758990275970262476, 6.19489569220104366113582384470, 7.02704569181704601844562747970, 7.51088355831290791095839984776, 9.158376545900250033237811636814, 9.577717426479420368779515890266

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