Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 2·13-s − 6·17-s − 8·19-s − 6·29-s + 4·31-s + 10·37-s + 6·41-s − 4·43-s + 49-s − 6·53-s − 12·59-s − 10·61-s − 4·67-s + 12·71-s + 10·73-s − 8·79-s − 12·83-s + 6·89-s − 2·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 6·119-s + ⋯
L(s)  = 1  + 0.377·7-s − 0.554·13-s − 1.45·17-s − 1.83·19-s − 1.11·29-s + 0.718·31-s + 1.64·37-s + 0.937·41-s − 0.609·43-s + 1/7·49-s − 0.824·53-s − 1.56·59-s − 1.28·61-s − 0.488·67-s + 1.42·71-s + 1.17·73-s − 0.900·79-s − 1.31·83-s + 0.635·89-s − 0.209·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.550·119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(25200\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{25200} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 25200,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.215252397\)
\(L(\frac12)\)  \(\approx\)  \(1.215252397\)
\(L(\frac{3}{2})\)   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 \( 1 \)
7 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−15.39855262461361, −14.82883654977721, −14.40462835436168, −13.73869676495468, −13.08903929228306, −12.80112508186360, −12.18854836771165, −11.43674217447401, −10.98405365883556, −10.67298135810389, −9.828478544290710, −9.280055454181626, −8.792407734812068, −8.108909568236756, −7.671886612196377, −6.909760238049787, −6.296026281498920, −5.914564510505979, −4.840686689490376, −4.506135083156855, −3.958838754659556, −2.902824684711581, −2.258647561777977, −1.676173480912871, −0.4166450202175092, 0.4166450202175092, 1.676173480912871, 2.258647561777977, 2.902824684711581, 3.958838754659556, 4.506135083156855, 4.840686689490376, 5.914564510505979, 6.296026281498920, 6.909760238049787, 7.671886612196377, 8.108909568236756, 8.792407734812068, 9.280055454181626, 9.828478544290710, 10.67298135810389, 10.98405365883556, 11.43674217447401, 12.18854836771165, 12.80112508186360, 13.08903929228306, 13.73869676495468, 14.40462835436168, 14.82883654977721, 15.39855262461361

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