Properties

Degree 2
Conductor $ 2^{4} \cdot 5^{2} \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  − 3·9-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 − 6·53-s − 4·59-s + 2·61-s + 8·67-s − 6·73-s + 9·81-s + 16·83-s + 6·89-s − 14·97-s + 12·99-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 9-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 − 0.824·53-s − 0.520·59-s + 0.256·61-s + 0.977·67-s − 0.702·73-s + 81-s + 1.75·83-s + 0.635·89-s − 1.42·97-s + 1.20·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(19600\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{19600} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 19600,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.003646014$
$L(\frac12)$  $\approx$  $1.003646014$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + p T^{2} \)
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

−15.77400040430666, −15.01137199649912, −14.64907803235418, −14.11710630055001, −13.46754905069330, −13.00864776649542, −12.37324041654874, −11.88230901075266, −11.14099384060926, −10.84627821748173, −10.13511449069469, −9.480065376340187, −9.043369169216928, −8.243215739790788, −7.783171768501600, −7.249257477674942, −6.540434283650715, −5.626915341617145, −5.286577496311584, −4.830666271522223, −3.644871897614838, −3.125621438534217, −2.501208244682031, −1.609655922575235, −0.4021213081468991, 0.4021213081468991, 1.609655922575235, 2.501208244682031, 3.125621438534217, 3.644871897614838, 4.830666271522223, 5.286577496311584, 5.626915341617145, 6.540434283650715, 7.249257477674942, 7.783171768501600, 8.243215739790788, 9.043369169216928, 9.480065376340187, 10.13511449069469, 10.84627821748173, 11.14099384060926, 11.88230901075266, 12.37324041654874, 13.00864776649542, 13.46754905069330, 14.11710630055001, 14.64907803235418, 15.01137199649912, 15.77400040430666

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