Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \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  + 3-s + 7-s + 9-s − 2·13-s + 6·17-s − 8·19-s + 21-s + 27-s + 6·29-s + 4·31-s + 10·37-s − 2·39-s − 6·41-s − 4·43-s + 49-s + 6·51-s + 6·53-s − 8·57-s + 12·59-s − 10·61-s + 63-s − 4·67-s − 12·71-s + 10·73-s − 8·79-s + 81-s + 12·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.554·13-s + 1.45·17-s − 1.83·19-s + 0.218·21-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 + 1/7·49-s + 0.840·51-s + 0.824·53-s − 1.05·57-s + 1.56·59-s − 1.28·61-s + 0.125·63-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8400} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 8400,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.712530060\)
\(L(\frac12)\)  \(\approx\)  \(2.712530060\)
\(L(\frac{3}{2})\)   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 \( 1 - T \)
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

−16.74108726401221, −16.56913154806856, −15.54246866012325, −15.09148909365829, −14.52841436082828, −14.22150305120158, −13.32603126664975, −12.94138079463094, −12.09436925620862, −11.82104163799331, −10.85961625360881, −10.16975563095571, −9.916031456602057, −8.950709428190540, −8.381626067445627, −7.922925278628559, −7.213735168276138, −6.447565995986516, −5.788089456279658, −4.801103708733275, −4.356553408121710, −3.409571979825148, −2.643218911540093, −1.876626856392683, −0.7924122026518692, 0.7924122026518692, 1.876626856392683, 2.643218911540093, 3.409571979825148, 4.356553408121710, 4.801103708733275, 5.788089456279658, 6.447565995986516, 7.213735168276138, 7.922925278628559, 8.381626067445627, 8.950709428190540, 9.916031456602057, 10.16975563095571, 10.85961625360881, 11.82104163799331, 12.09436925620862, 12.94138079463094, 13.32603126664975, 14.22150305120158, 14.52841436082828, 15.09148909365829, 15.54246866012325, 16.56913154806856, 16.74108726401221

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