Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
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 − 7-s + 9-s − 6·11-s − 2·13-s − 4·19-s + 21-s + 6·23-s − 27-s + 6·29-s + 8·31-s + 6·33-s − 2·37-s + 2·39-s + 12·41-s + 4·43-s − 12·47-s + 49-s + 6·53-s + 4·57-s − 10·61-s − 63-s − 8·67-s − 6·69-s + 6·71-s + 10·73-s + 6·77-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.80·11-s − 0.554·13-s − 0.917·19-s + 0.218·21-s + 1.25·23-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 1.04·33-s − 0.328·37-s + 0.320·39-s + 1.87·41-s + 0.609·43-s − 1.75·47-s + 1/7·49-s + 0.824·53-s + 0.529·57-s − 1.28·61-s − 0.125·63-s − 0.977·67-s − 0.722·69-s + 0.712·71-s + 1.17·73-s + 0.683·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{2100} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2100,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.9795395896\)
\(L(\frac12)\)  \(\approx\)  \(0.9795395896\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−9.160755535964379354523489733441, −8.208792959356633127272389160552, −7.56907459997685032113873998444, −6.68428738506159861567855023989, −5.96686399911741876769524513282, −5.00463732145876448988227255567, −4.53602693910607323330756743987, −3.08167105333254803920115617344, −2.35301068723521734945649298411, −0.64563828008458494707754753873, 0.64563828008458494707754753873, 2.35301068723521734945649298411, 3.08167105333254803920115617344, 4.53602693910607323330756743987, 5.00463732145876448988227255567, 5.96686399911741876769524513282, 6.68428738506159861567855023989, 7.56907459997685032113873998444, 8.208792959356633127272389160552, 9.160755535964379354523489733441

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