Properties

Degree 2
Conductor $ 2^{6} \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

\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(33600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{33600} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 33600,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.232561212\)
\(L(\frac12)\)  \(\approx\)  \(2.232561212\)
\(L(\frac{3}{2})\)   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

−14.95785187191935, −14.56293093687707, −13.94121740365507, −13.44852828725361, −12.83833516108726, −12.55873418524206, −11.91537209503676, −11.24428450272660, −10.58635470883197, −10.26873622007054, −9.560345284440666, −9.117582368142585, −8.354136765551018, −8.184944846787444, −7.380024400428196, −6.724250106080020, −6.353631651853751, −5.409410938383966, −5.127804915829586, −3.916734554350276, −3.753503538885082, −3.079671759691536, −2.110657816294656, −1.664010788551633, −0.5282401016773331, 0.5282401016773331, 1.664010788551633, 2.110657816294656, 3.079671759691536, 3.753503538885082, 3.916734554350276, 5.127804915829586, 5.409410938383966, 6.353631651853751, 6.724250106080020, 7.380024400428196, 8.184944846787444, 8.354136765551018, 9.117582368142585, 9.560345284440666, 10.26873622007054, 10.58635470883197, 11.24428450272660, 11.91537209503676, 12.55873418524206, 12.83833516108726, 13.44852828725361, 13.94121740365507, 14.56293093687707, 14.95785187191935

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