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 − 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  =  \(0\)
Selberg data  =  \((2,\ 2100,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.422482730\)
\(L(\frac12)\)  \(\approx\)  \(2.422482730\)
\(L(\frac{3}{2})\)   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

−8.997461625279297363034935811227, −8.358505979296918632239518392800, −7.61848870762004193638309186711, −6.95638146741475340569582445723, −5.86120657998933401256717519917, −5.11430793144295736663953593583, −4.09854389466643786070253476477, −3.25661912312057055898711123454, −2.26065795162526057830993395440, −1.06458588106369142400126897748, 1.06458588106369142400126897748, 2.26065795162526057830993395440, 3.25661912312057055898711123454, 4.09854389466643786070253476477, 5.11430793144295736663953593583, 5.86120657998933401256717519917, 6.95638146741475340569582445723, 7.61848870762004193638309186711, 8.358505979296918632239518392800, 8.997461625279297363034935811227

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