Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 11-s − 2·13-s + 4·17-s − 6·19-s − 8·29-s + 8·31-s + 10·37-s − 8·41-s + 2·43-s + 8·47-s − 3·49-s + 2·53-s − 12·59-s − 10·61-s − 12·67-s + 8·71-s − 6·73-s + 2·77-s + 2·79-s + 16·83-s + 14·89-s + 4·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.755·7-s − 0.301·11-s − 0.554·13-s + 0.970·17-s − 1.37·19-s − 1.48·29-s + 1.43·31-s + 1.64·37-s − 1.24·41-s + 0.304·43-s + 1.16·47-s − 3/7·49-s + 0.274·53-s − 1.56·59-s − 1.28·61-s − 1.46·67-s + 0.949·71-s − 0.702·73-s + 0.227·77-s + 0.225·79-s + 1.75·83-s + 1.48·89-s + 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(158400\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{158400} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 158400,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
11 \( 1 + T \)
good7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.47350253754596, −13.07288031660504, −12.51481072212811, −12.21691998926083, −11.73280143289487, −11.09393806247876, −10.55815210657735, −10.23150083180377, −9.725632381955419, −9.186705466431305, −8.895873139562659, −7.978747088594795, −7.826476387483468, −7.279849343719170, −6.538602295964479, −6.192854914035897, −5.773550013640517, −5.043152082007297, −4.542492816842399, −4.008951617451946, −3.327245833598506, −2.856803073074107, −2.262327452331401, −1.593029981166796, −0.6977870324441098, 0, 0.6977870324441098, 1.593029981166796, 2.262327452331401, 2.856803073074107, 3.327245833598506, 4.008951617451946, 4.542492816842399, 5.043152082007297, 5.773550013640517, 6.192854914035897, 6.538602295964479, 7.279849343719170, 7.826476387483468, 7.978747088594795, 8.895873139562659, 9.186705466431305, 9.725632381955419, 10.23150083180377, 10.55815210657735, 11.09393806247876, 11.73280143289487, 12.21691998926083, 12.51481072212811, 13.07288031660504, 13.47350253754596

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