Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 4·11-s + 2·13-s + 2·17-s + 4·19-s + 4·23-s + 25-s + 2·29-s − 8·31-s + 6·37-s − 6·41-s + 8·43-s − 4·47-s − 6·53-s + 4·55-s + 4·59-s + 2·61-s + 2·65-s − 8·67-s + 6·73-s + 16·83-s + 2·85-s − 6·89-s + 4·95-s + 14·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.986·37-s − 0.937·41-s + 1.21·43-s − 0.583·47-s − 0.824·53-s + 0.539·55-s + 0.520·59-s + 0.256·61-s + 0.248·65-s − 0.977·67-s + 0.702·73-s + 1.75·83-s + 0.216·85-s − 0.635·89-s + 0.410·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(35280\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{35280} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 35280,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.524490948$
$L(\frac12)$  $\approx$  $3.524490948$
$L(\frac{3}{2})$   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 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
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

−14.73802754081346, −14.44359743283061, −13.98784164950440, −13.32219125108185, −12.95187841888325, −12.26766472604867, −11.78567750930786, −11.22357386459491, −10.79862094750859, −10.07611564684429, −9.494695733892280, −9.131709774722811, −8.644534324359050, −7.822217156979086, −7.345430388461787, −6.645311368613738, −6.204565216899671, −5.553417691938207, −4.999646167073406, −4.249874490593700, −3.522064829754538, −3.104690799660728, −2.103662175729141, −1.371356847318043, −0.7724438171067429, 0.7724438171067429, 1.371356847318043, 2.103662175729141, 3.104690799660728, 3.522064829754538, 4.249874490593700, 4.999646167073406, 5.553417691938207, 6.204565216899671, 6.645311368613738, 7.345430388461787, 7.822217156979086, 8.644534324359050, 9.131709774722811, 9.494695733892280, 10.07611564684429, 10.79862094750859, 11.22357386459491, 11.78567750930786, 12.26766472604867, 12.95187841888325, 13.32219125108185, 13.98784164950440, 14.44359743283061, 14.73802754081346

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