Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5^{2} $
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 − 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

\( d \)  =  \(2\)
\( N \)  =  \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1200} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1200,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.046299298\)
\(L(\frac12)\)  \(\approx\)  \(1.046299298\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.882613299713563149760910871277, −9.250912384445108460730408413088, −7.66165741262556570263007689206, −7.55015404230883423512821682767, −6.16960063995750489450981625407, −5.71322079987385705135293329202, −4.69169079872189797958426439923, −3.50574880723700274234440164435, −2.58093474510021419032571611233, −0.77604421210661452476716259489, 0.77604421210661452476716259489, 2.58093474510021419032571611233, 3.50574880723700274234440164435, 4.69169079872189797958426439923, 5.71322079987385705135293329202, 6.16960063995750489450981625407, 7.55015404230883423512821682767, 7.66165741262556570263007689206, 9.250912384445108460730408413088, 9.882613299713563149760910871277

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