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 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 6·11-s − 2·13-s + 2·17-s − 4·19-s + 4·23-s − 2·29-s − 2·31-s + 10·37-s + 6·41-s + 2·43-s − 2·47-s + 49-s + 6·53-s + 4·59-s + 12·61-s − 10·67-s + 12·71-s − 2·73-s + 6·77-s − 16·79-s − 12·83-s + 14·89-s + 2·91-s + 18·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.80·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s + 0.834·23-s − 0.371·29-s − 0.359·31-s + 1.64·37-s + 0.937·41-s + 0.304·43-s − 0.291·47-s + 1/7·49-s + 0.824·53-s + 0.520·59-s + 1.53·61-s − 1.22·67-s + 1.42·71-s − 0.234·73-s + 0.683·77-s − 1.80·79-s − 1.31·83-s + 1.48·89-s + 0.209·91-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-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  =  \(0\)
Selberg data  =  \((2,\ 100800,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.101741352\)
\(L(\frac12)\)  \(\approx\)  \(1.101741352\)
\(L(\frac{3}{2})\)   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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.42409319451461, −13.29405603640162, −12.74866848818180, −12.55122571565964, −11.76945500673847, −11.20558615385949, −10.80329467698637, −10.24888855616190, −9.931986404624457, −9.312037210132633, −8.809278310154347, −8.140259933371632, −7.754931457144318, −7.295509919335074, −6.731029046011633, −6.058045748874698, −5.470847580180836, −5.185656158968111, −4.416484631170081, −3.945184883444396, −3.082374967856086, −2.572299664711897, −2.248527196154970, −1.155334848835892, −0.3470508090578708, 0.3470508090578708, 1.155334848835892, 2.248527196154970, 2.572299664711897, 3.082374967856086, 3.945184883444396, 4.416484631170081, 5.185656158968111, 5.470847580180836, 6.058045748874698, 6.731029046011633, 7.295509919335074, 7.754931457144318, 8.140259933371632, 8.809278310154347, 9.312037210132633, 9.931986404624457, 10.24888855616190, 10.80329467698637, 11.20558615385949, 11.76945500673847, 12.55122571565964, 12.74866848818180, 13.29405603640162, 13.42409319451461

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