Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 2·13-s − 6·17-s − 8·19-s + 27-s − 6·29-s − 4·31-s − 10·37-s − 2·39-s + 6·41-s + 4·43-s − 6·51-s − 6·53-s − 8·57-s + 12·59-s − 10·61-s + 4·67-s − 12·71-s − 10·73-s − 8·79-s + 81-s + 12·83-s − 6·87-s + 6·89-s − 4·93-s − 10·97-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.554·13-s − 1.45·17-s − 1.83·19-s + 0.192·27-s − 1.11·29-s − 0.718·31-s − 1.64·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.840·51-s − 0.824·53-s − 1.05·57-s + 1.56·59-s − 1.28·61-s + 0.488·67-s − 1.42·71-s − 1.17·73-s − 0.900·79-s + 1/9·81-s + 1.31·83-s − 0.643·87-s + 0.635·89-s − 0.414·93-s − 1.01·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(235200\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{235200} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(2\)
Selberg data  =  \((2,\ 235200,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.39182472169175, −12.98075227159389, −12.47904392354552, −12.16093473023371, −11.43181763635467, −10.93028731629939, −10.65489357943359, −10.20746505442130, −9.388516738917597, −9.237244651703120, −8.658753507442678, −8.372451796469225, −7.688385225097239, −7.215404268717093, −6.816974531788808, −6.295682850669530, −5.737075948726264, −5.136348808087400, −4.480614623870300, −4.161115211791177, −3.656346806832715, −2.914317150152996, −2.288018242282653, −2.008495125104859, −1.329078496122330, 0, 0, 1.329078496122330, 2.008495125104859, 2.288018242282653, 2.914317150152996, 3.656346806832715, 4.161115211791177, 4.480614623870300, 5.136348808087400, 5.737075948726264, 6.295682850669530, 6.816974531788808, 7.215404268717093, 7.688385225097239, 8.372451796469225, 8.658753507442678, 9.237244651703120, 9.388516738917597, 10.20746505442130, 10.65489357943359, 10.93028731629939, 11.43181763635467, 12.16093473023371, 12.47904392354552, 12.98075227159389, 13.39182472169175

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