Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s − 11-s + 4·13-s − 6·17-s + 4·19-s − 6·23-s − 6·29-s − 8·31-s + 10·37-s − 6·41-s + 8·43-s + 6·47-s − 3·49-s + 8·61-s − 4·67-s + 6·71-s − 2·73-s − 2·77-s − 14·79-s + 12·83-s + 6·89-s + 8·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.755·7-s − 0.301·11-s + 1.10·13-s − 1.45·17-s + 0.917·19-s − 1.25·23-s − 1.11·29-s − 1.43·31-s + 1.64·37-s − 0.937·41-s + 1.21·43-s + 0.875·47-s − 3/7·49-s + 1.02·61-s − 0.488·67-s + 0.712·71-s − 0.234·73-s − 0.227·77-s − 1.57·79-s + 1.31·83-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(39600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{39600} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 39600,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
11 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−15.04111020581960, −14.57993555159381, −13.88621184960818, −13.62105574748988, −12.96263080867888, −12.61979714885493, −11.68742135431238, −11.38306329816320, −10.97296645008486, −10.48205742890030, −9.673125458728039, −9.218365526257102, −8.652176522049526, −8.113902947769451, −7.572168135123871, −7.077084385290768, −6.257418872403367, −5.771700968479560, −5.262546136760025, −4.428162418792689, −3.997523063636244, −3.349543155950314, −2.359116587421795, −1.883572502230438, −1.052547587859108, 0, 1.052547587859108, 1.883572502230438, 2.359116587421795, 3.349543155950314, 3.997523063636244, 4.428162418792689, 5.262546136760025, 5.771700968479560, 6.257418872403367, 7.077084385290768, 7.572168135123871, 8.113902947769451, 8.652176522049526, 9.218365526257102, 9.673125458728039, 10.48205742890030, 10.97296645008486, 11.38306329816320, 11.68742135431238, 12.61979714885493, 12.96263080867888, 13.62105574748988, 13.88621184960818, 14.57993555159381, 15.04111020581960

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