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 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 11-s − 12-s − 7·13-s + 16-s + 4·17-s − 18-s + 19-s + 22-s − 23-s + 24-s + 7·26-s − 27-s − 8·29-s + 6·31-s − 32-s + 33-s − 4·34-s + 36-s + 3·37-s − 38-s + 7·39-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 − 0.301·11-s − 0.288·12-s − 1.94·13-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.229·19-s + 0.213·22-s − 0.208·23-s + 0.204·24-s + 1.37·26-s − 0.192·27-s − 1.48·29-s + 1.07·31-s − 0.176·32-s + 0.174·33-s − 0.685·34-s + 1/6·36-s + 0.493·37-s − 0.162·38-s + 1.12·39-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  =  \(1\)
Selberg data  =  \((2,\ 7350,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 + T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−7.49082035325798808627589780843, −7.17859849879754654403995664249, −6.12743460060459553474267724678, −5.55929255512570164244895593900, −4.85906600486342774120768515337, −4.01264780578647336422952714188, −2.86462309925535509293443510036, −2.21471342708607028514697493587, −1.03446535266972550117746122432, 0, 1.03446535266972550117746122432, 2.21471342708607028514697493587, 2.86462309925535509293443510036, 4.01264780578647336422952714188, 4.85906600486342774120768515337, 5.55929255512570164244895593900, 6.12743460060459553474267724678, 7.17859849879754654403995664249, 7.49082035325798808627589780843

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