Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} \cdot 7 $
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 + 7-s − 8-s + 9-s + 2·11-s − 12-s − 13-s − 14-s + 16-s + 17-s − 18-s + 4·19-s − 21-s − 2·22-s − 7·23-s + 24-s + 26-s − 27-s + 28-s + 29-s + 3·31-s − 32-s − 2·33-s − 34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.218·21-s − 0.426·22-s − 1.45·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + 0.185·29-s + 0.538·31-s − 0.176·32-s − 0.348·33-s − 0.171·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1050} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1050,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.002711243$
$L(\frac12)$  $\approx$  $1.002711243$
$L(\frac{3}{2})$   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 \( 1 - T \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−9.867884616148654764319249926263, −9.260468983053516807245881636264, −8.158214749417890256597687504313, −7.55252193275581649651700207024, −6.56557884551623356952409312838, −5.80952482446894196152644579304, −4.77770790573633599440232036463, −3.65078540287386924344518852742, −2.19259007680211571616293034668, −0.907613785551433006810129340306, 0.907613785551433006810129340306, 2.19259007680211571616293034668, 3.65078540287386924344518852742, 4.77770790573633599440232036463, 5.80952482446894196152644579304, 6.56557884551623356952409312838, 7.55252193275581649651700207024, 8.158214749417890256597687504313, 9.260468983053516807245881636264, 9.867884616148654764319249926263

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