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 − 6·13-s + 2·17-s + 8·19-s + 8·23-s − 2·29-s + 4·31-s − 2·37-s + 6·41-s + 4·43-s + 8·47-s + 49-s − 10·53-s + 4·59-s + 2·61-s + 4·67-s + 12·71-s + 2·73-s + 8·79-s + 4·83-s + 6·89-s + 6·91-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.66·13-s + 0.485·17-s + 1.83·19-s + 1.66·23-s − 0.371·29-s + 0.718·31-s − 0.328·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 1.37·53-s + 0.520·59-s + 0.256·61-s + 0.488·67-s + 1.42·71-s + 0.234·73-s + 0.900·79-s + 0.439·83-s + 0.635·89-s + 0.628·91-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

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

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  =  $(2,\ 100800,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.777274232$
$L(\frac12)$  $\approx$  $2.777274232$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.80862564973686, −13.22474963402418, −12.67891431367522, −12.31671699049049, −11.88706543266123, −11.32333857594726, −10.83004289406527, −10.22373826560904, −9.664877255146692, −9.413142164677854, −8.994332723105635, −8.134383148265220, −7.599528595672463, −7.273215009082166, −6.837155589899355, −6.093114220547402, −5.475309487965591, −5.000371468317004, −4.665539116442852, −3.713265901050085, −3.217017703284143, −2.672808653572593, −2.114792124481348, −1.033494391099619, −0.6241286256797853, 0.6241286256797853, 1.033494391099619, 2.114792124481348, 2.672808653572593, 3.217017703284143, 3.713265901050085, 4.665539116442852, 5.000371468317004, 5.475309487965591, 6.093114220547402, 6.837155589899355, 7.273215009082166, 7.599528595672463, 8.134383148265220, 8.994332723105635, 9.413142164677854, 9.664877255146692, 10.22373826560904, 10.83004289406527, 11.32333857594726, 11.88706543266123, 12.31671699049049, 12.67891431367522, 13.22474963402418, 13.80862564973686

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