Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17 $
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 − 4·7-s + 9-s + 4·11-s − 13-s − 17-s − 8·19-s + 4·21-s − 27-s − 2·29-s − 4·33-s + 10·37-s + 39-s + 10·41-s − 8·43-s + 9·49-s + 51-s − 6·53-s + 8·57-s + 8·59-s + 14·61-s − 4·63-s − 4·67-s + 8·71-s + 6·73-s − 16·77-s + 4·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.242·17-s − 1.83·19-s + 0.872·21-s − 0.192·27-s − 0.371·29-s − 0.696·33-s + 1.64·37-s + 0.160·39-s + 1.56·41-s − 1.21·43-s + 9/7·49-s + 0.140·51-s − 0.824·53-s + 1.05·57-s + 1.04·59-s + 1.79·61-s − 0.503·63-s − 0.488·67-s + 0.949·71-s + 0.702·73-s − 1.82·77-s + 0.450·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(265200\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{265200} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 265200,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.173186954$
$L(\frac12)$  $\approx$  $1.173186954$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−12.70278427801761, −12.47771416778450, −11.92150254803620, −11.34166873347079, −11.06512143248047, −10.50782245423058, −9.866104570286672, −9.710018332181685, −9.143597696747124, −8.787117381021653, −8.101938377043321, −7.616169941830722, −6.792697696475040, −6.555901672866440, −6.419453082968214, −5.747619221665675, −5.262597604013642, −4.412363388006993, −4.089003665180022, −3.724917029711294, −2.948278250630313, −2.395999098150488, −1.811291548157965, −0.9188210962621777, −0.3676060917996702, 0.3676060917996702, 0.9188210962621777, 1.811291548157965, 2.395999098150488, 2.948278250630313, 3.724917029711294, 4.089003665180022, 4.412363388006993, 5.262597604013642, 5.747619221665675, 6.419453082968214, 6.555901672866440, 6.792697696475040, 7.616169941830722, 8.101938377043321, 8.787117381021653, 9.143597696747124, 9.710018332181685, 9.866104570286672, 10.50782245423058, 11.06512143248047, 11.34166873347079, 11.92150254803620, 12.47771416778450, 12.70278427801761

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