Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 7^{2} \cdot 11^{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-s − 2·5-s + 9-s − 2·13-s + 2·15-s + 4·17-s − 6·19-s − 25-s − 27-s + 8·29-s + 8·31-s + 10·37-s + 2·39-s + 8·41-s + 2·43-s − 2·45-s + 8·47-s − 4·51-s − 2·53-s + 6·57-s − 12·59-s + 10·61-s + 4·65-s + 12·67-s + 8·71-s + 6·73-s + 75-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.554·13-s + 0.516·15-s + 0.970·17-s − 1.37·19-s − 1/5·25-s − 0.192·27-s + 1.48·29-s + 1.43·31-s + 1.64·37-s + 0.320·39-s + 1.24·41-s + 0.304·43-s − 0.298·45-s + 1.16·47-s − 0.560·51-s − 0.274·53-s + 0.794·57-s − 1.56·59-s + 1.28·61-s + 0.496·65-s + 1.46·67-s + 0.949·71-s + 0.702·73-s + 0.115·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(71148\)    =    \(2^{2} \cdot 3 \cdot 7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{71148} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 71148,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.816300812$
$L(\frac12)$  $\approx$  $1.816300812$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
7 \( 1 \)
11 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−13.98822703728366, −13.80963030543177, −12.82547939709279, −12.56281318247903, −12.10122247164192, −11.73790515561263, −11.06281172967207, −10.74488571209141, −10.11496711686129, −9.671205546836321, −9.095080791649986, −8.227248275008652, −8.011987628030446, −7.543356942718249, −6.791931595132818, −6.316393422631691, −5.876786803493968, −5.064957027754504, −4.525858512166730, −4.159184413052363, −3.484323360332577, −2.649286579199712, −2.175684971930584, −0.9187346286767975, −0.6160392343013462, 0.6160392343013462, 0.9187346286767975, 2.175684971930584, 2.649286579199712, 3.484323360332577, 4.159184413052363, 4.525858512166730, 5.064957027754504, 5.876786803493968, 6.316393422631691, 6.791931595132818, 7.543356942718249, 8.011987628030446, 8.227248275008652, 9.095080791649986, 9.671205546836321, 10.11496711686129, 10.74488571209141, 11.06281172967207, 11.73790515561263, 12.10122247164192, 12.56281318247903, 12.82547939709279, 13.80963030543177, 13.98822703728366

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