Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 5^{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  + 4·7-s + 4·11-s − 2·13-s + 2·17-s − 4·19-s + 4·23-s − 2·29-s − 8·31-s + 6·37-s + 6·41-s − 8·43-s + 4·47-s + 9·49-s − 6·53-s − 4·59-s + 2·61-s + 8·67-s + 6·73-s + 16·77-s + 16·83-s + 6·89-s − 8·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 1.51·7-s + 1.20·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s + 0.834·23-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 + 9/7·49-s − 0.824·53-s − 0.520·59-s + 0.256·61-s + 0.977·67-s + 0.702·73-s + 1.82·77-s + 1.75·83-s + 0.635·89-s − 0.838·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(14400\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{14400} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 14400,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.948800690$
$L(\frac12)$  $\approx$  $2.948800690$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
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

−16.30476658886431, −15.29485882259204, −14.79875348887469, −14.59331182492602, −14.10966064653918, −13.33055379570878, −12.63890998350501, −12.18149898483082, −11.44392503880085, −11.13728641037175, −10.60587160725058, −9.716724368769951, −9.159867881122416, −8.665397323902617, −7.897109332237044, −7.498418345689588, −6.744720801505031, −6.074004025911819, −5.258363160667637, −4.743440595376873, −4.085990382585352, −3.377551132315697, −2.238692883379342, −1.679867704714602, −0.7906555535471514, 0.7906555535471514, 1.679867704714602, 2.238692883379342, 3.377551132315697, 4.085990382585352, 4.743440595376873, 5.258363160667637, 6.074004025911819, 6.744720801505031, 7.498418345689588, 7.897109332237044, 8.665397323902617, 9.159867881122416, 9.716724368769951, 10.60587160725058, 11.13728641037175, 11.44392503880085, 12.18149898483082, 12.63890998350501, 13.33055379570878, 14.10966064653918, 14.59331182492602, 14.79875348887469, 15.29485882259204, 16.30476658886431

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