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  − 3·3-s + 6·9-s + 11-s + 2·13-s + 3·17-s + 5·19-s − 3·23-s − 9·27-s − 6·29-s − 31-s − 3·33-s + 5·37-s − 6·39-s + 10·41-s − 4·43-s − 47-s − 9·51-s + 9·53-s − 15·57-s + 3·59-s − 3·61-s + 11·67-s + 9·69-s − 16·71-s + 7·73-s + 11·79-s + 9·81-s + ⋯
L(s)  = 1  − 1.73·3-s + 2·9-s + 0.301·11-s + 0.554·13-s + 0.727·17-s + 1.14·19-s − 0.625·23-s − 1.73·27-s − 1.11·29-s − 0.179·31-s − 0.522·33-s + 0.821·37-s − 0.960·39-s + 1.56·41-s − 0.609·43-s − 0.145·47-s − 1.26·51-s + 1.23·53-s − 1.98·57-s + 0.390·59-s − 0.384·61-s + 1.34·67-s + 1.08·69-s − 1.89·71-s + 0.819·73-s + 1.23·79-s + 81-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  =  $(2,\ 19600,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.314273617$
$L(\frac12)$  $\approx$  $1.314273617$
$L(\frac{3}{2})$   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 \( 1 + p T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−15.92026975410728, −15.32990815831051, −14.60588340883713, −14.07525438416158, −13.24366559772656, −12.94175553862397, −12.13016307769938, −11.84341821393693, −11.33280519384736, −10.83397048389655, −10.26452800972355, −9.645939309163899, −9.214685609500232, −8.218208760570322, −7.511670071476873, −7.120449672738354, −6.228751575710689, −5.909241099256574, −5.369834917896218, −4.741458416627420, −3.962854074338414, −3.395914987402821, −2.157880851694797, −1.202177087494433, −0.6184840073684046, 0.6184840073684046, 1.202177087494433, 2.157880851694797, 3.395914987402821, 3.962854074338414, 4.741458416627420, 5.369834917896218, 5.909241099256574, 6.228751575710689, 7.120449672738354, 7.511670071476873, 8.218208760570322, 9.214685609500232, 9.645939309163899, 10.26452800972355, 10.83397048389655, 11.33280519384736, 11.84341821393693, 12.13016307769938, 12.94175553862397, 13.24366559772656, 14.07525438416158, 14.60588340883713, 15.32990815831051, 15.92026975410728

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