Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \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  − 7-s − 4·11-s − 2·13-s − 6·17-s − 6·29-s − 6·37-s + 10·41-s + 12·47-s + 49-s + 6·53-s + 4·59-s − 2·61-s − 8·67-s − 4·71-s − 10·73-s + 4·77-s + 12·83-s + 10·89-s + 2·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 6·119-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.20·11-s − 0.554·13-s − 1.45·17-s − 1.11·29-s − 0.986·37-s + 1.56·41-s + 1.75·47-s + 1/7·49-s + 0.824·53-s + 0.520·59-s − 0.256·61-s − 0.977·67-s − 0.474·71-s − 1.17·73-s + 0.455·77-s + 1.31·83-s + 1.05·89-s + 0.209·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.550·119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(100800\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{100800} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 100800,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.85135878557489, −13.42385373432304, −12.99140195809831, −12.73837038187292, −11.91973615278503, −11.71279361588196, −10.76983306833739, −10.67814807455106, −10.20546846493091, −9.424172832974061, −9.047683071396936, −8.679225164177102, −7.858441377771180, −7.441374511328177, −7.098061720806715, −6.357532583663192, −5.837184748357358, −5.319708322547202, −4.750852710393199, −4.165626498590165, −3.615383497270133, −2.737919347904118, −2.418307111930656, −1.794992337476674, −0.6667432600571413, 0, 0.6667432600571413, 1.794992337476674, 2.418307111930656, 2.737919347904118, 3.615383497270133, 4.165626498590165, 4.750852710393199, 5.319708322547202, 5.837184748357358, 6.357532583663192, 7.098061720806715, 7.441374511328177, 7.858441377771180, 8.679225164177102, 9.047683071396936, 9.424172832974061, 10.20546846493091, 10.67814807455106, 10.76983306833739, 11.71279361588196, 11.91973615278503, 12.73837038187292, 12.99140195809831, 13.42385373432304, 13.85135878557489

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