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 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s + 11-s + 2·13-s − 2·17-s − 8·23-s + 6·29-s + 8·31-s − 6·37-s + 2·41-s − 8·47-s + 9·49-s + 6·53-s − 4·59-s + 6·61-s − 4·67-s + 14·73-s + 4·77-s + 4·79-s − 12·83-s + 6·89-s + 8·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.51·7-s + 0.301·11-s + 0.554·13-s − 0.485·17-s − 1.66·23-s + 1.11·29-s + 1.43·31-s − 0.986·37-s + 0.312·41-s − 1.16·47-s + 9/7·49-s + 0.824·53-s − 0.520·59-s + 0.768·61-s − 0.488·67-s + 1.63·73-s + 0.455·77-s + 0.450·79-s − 1.31·83-s + 0.635·89-s + 0.838·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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  =  0
Selberg data  =  $(2,\ 39600,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.087394211$
$L(\frac12)$  $\approx$  $3.087394211$
$L(\frac{3}{2})$   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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.67862288612533, −14.23972131434874, −13.82220676064146, −13.48365921903929, −12.59786088881805, −12.07073240428379, −11.63168265594292, −11.27100953651105, −10.55371567563995, −10.18787919618056, −9.541429175012913, −8.700401083909989, −8.381038051116940, −7.996529230341612, −7.348590543526686, −6.526627487093159, −6.207007548826161, −5.363862754717422, −4.833229306905163, −4.274296316908768, −3.754540741694603, −2.808830358726254, −2.041305463110971, −1.506231319955804, −0.6627472778246820, 0.6627472778246820, 1.506231319955804, 2.041305463110971, 2.808830358726254, 3.754540741694603, 4.274296316908768, 4.833229306905163, 5.363862754717422, 6.207007548826161, 6.526627487093159, 7.348590543526686, 7.996529230341612, 8.381038051116940, 8.700401083909989, 9.541429175012913, 10.18787919618056, 10.55371567563995, 11.27100953651105, 11.63168265594292, 12.07073240428379, 12.59786088881805, 13.48365921903929, 13.82220676064146, 14.23972131434874, 14.67862288612533

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