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 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 7-s + 9-s − 11-s − 2·13-s + 6·19-s + 21-s + 23-s − 27-s + 29-s − 2·31-s + 33-s − 7·37-s + 2·39-s − 8·41-s − 43-s − 2·47-s + 49-s − 14·53-s − 6·57-s + 10·59-s − 63-s − 3·67-s − 69-s − 9·71-s + 77-s + 79-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 1.37·19-s + 0.218·21-s + 0.208·23-s − 0.192·27-s + 0.185·29-s − 0.359·31-s + 0.174·33-s − 1.15·37-s + 0.320·39-s − 1.24·41-s − 0.152·43-s − 0.291·47-s + 1/7·49-s − 1.92·53-s − 0.794·57-s + 1.30·59-s − 0.125·63-s − 0.366·67-s − 0.120·69-s − 1.06·71-s + 0.113·77-s + 0.112·79-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  =  \(1\)
Selberg data  =  \((2,\ 2100,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 + T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−8.781529339256990613587869029567, −7.80298801990183034032490335907, −7.10913822752986093173591589044, −6.40155300612741441209080116015, −5.37994560739225437445244358999, −4.93189704452359005958027193915, −3.71059370521019590152341252906, −2.82842343595315456135468530958, −1.47588350858440766249674578424, 0, 1.47588350858440766249674578424, 2.82842343595315456135468530958, 3.71059370521019590152341252906, 4.93189704452359005958027193915, 5.37994560739225437445244358999, 6.40155300612741441209080116015, 7.10913822752986093173591589044, 7.80298801990183034032490335907, 8.781529339256990613587869029567

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