Properties

Degree 2
Conductor $ 3^{2} \cdot 5^{2} \cdot 7 \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  + 2-s − 4-s + 7-s − 3·8-s + 6·13-s + 14-s − 16-s − 2·17-s − 4·19-s + 6·26-s − 28-s − 2·29-s + 8·31-s + 5·32-s − 2·34-s − 6·37-s − 4·38-s + 10·41-s − 4·43-s − 8·47-s + 49-s − 6·52-s + 6·53-s − 3·56-s − 2·58-s − 4·59-s + 10·61-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s + 1.66·13-s + 0.267·14-s − 1/4·16-s − 0.485·17-s − 0.917·19-s + 1.17·26-s − 0.188·28-s − 0.371·29-s + 1.43·31-s + 0.883·32-s − 0.342·34-s − 0.986·37-s − 0.648·38-s + 1.56·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.832·52-s + 0.824·53-s − 0.400·56-s − 0.262·58-s − 0.520·59-s + 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(190575\)    =    \(3^{2} \cdot 5^{2} \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{190575} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 190575,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.901192091$
$L(\frac12)$  $\approx$  $2.901192091$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good2 \( 1 - T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−13.08864137230128, −12.75629747642137, −12.40675169911174, −11.52832816996748, −11.40817740195166, −10.95514704097447, −10.16931325189279, −9.972666010021363, −9.167346940518820, −8.738325902026065, −8.372467750564006, −8.120999930558303, −7.226032738012645, −6.616594175022908, −6.267719634523544, −5.726496464761837, −5.302140345348989, −4.584464594189375, −4.255748000749817, −3.773239998898003, −3.234579439771950, −2.588697066070746, −1.889660361296940, −1.157869521450271, −0.4677671949628428, 0.4677671949628428, 1.157869521450271, 1.889660361296940, 2.588697066070746, 3.234579439771950, 3.773239998898003, 4.255748000749817, 4.584464594189375, 5.302140345348989, 5.726496464761837, 6.267719634523544, 6.616594175022908, 7.226032738012645, 8.120999930558303, 8.372467750564006, 8.738325902026065, 9.167346940518820, 9.972666010021363, 10.16931325189279, 10.95514704097447, 11.40817740195166, 11.52832816996748, 12.40675169911174, 12.75629747642137, 13.08864137230128

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