Properties

Degree 2
Conductor $ 2 \cdot 3 \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  − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 5·11-s + 12-s + 16-s + 4·17-s − 18-s + 8·19-s − 5·22-s + 4·23-s − 24-s + 27-s − 5·29-s + 3·31-s − 32-s + 5·33-s − 4·34-s + 36-s + 4·37-s − 8·38-s − 2·43-s + 5·44-s − 4·46-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.50·11-s + 0.288·12-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 1.83·19-s − 1.06·22-s + 0.834·23-s − 0.204·24-s + 0.192·27-s − 0.928·29-s + 0.538·31-s − 0.176·32-s + 0.870·33-s − 0.685·34-s + 1/6·36-s + 0.657·37-s − 1.29·38-s − 0.304·43-s + 0.753·44-s − 0.589·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{7350} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 7350,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.452519931\)
\(L(\frac12)\)  \(\approx\)  \(2.452519931\)
\(L(\frac{3}{2})\)   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 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−7.81967241498152900815704220628, −7.40759340662978289434037906130, −6.74693828644844264667222732489, −5.92833840279769606874916637934, −5.18371661920317724416864995160, −4.11308576803390779259894475306, −3.39928963086299127374750424516, −2.73953911727052120787343823893, −1.50567632180377324909373109770, −0.973457831401776018640109900133, 0.973457831401776018640109900133, 1.50567632180377324909373109770, 2.73953911727052120787343823893, 3.39928963086299127374750424516, 4.11308576803390779259894475306, 5.18371661920317724416864995160, 5.92833840279769606874916637934, 6.74693828644844264667222732489, 7.40759340662978289434037906130, 7.81967241498152900815704220628

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