Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \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 + 4-s − 7-s − 8-s − 6·11-s + 13-s + 14-s + 16-s + 3·17-s − 4·19-s + 6·22-s − 3·23-s − 26-s − 28-s − 3·29-s + 5·31-s − 32-s − 3·34-s + 10·37-s + 4·38-s − 9·41-s + 43-s − 6·44-s + 3·46-s + 49-s + 52-s + 9·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.80·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.917·19-s + 1.27·22-s − 0.625·23-s − 0.196·26-s − 0.188·28-s − 0.557·29-s + 0.898·31-s − 0.176·32-s − 0.514·34-s + 1.64·37-s + 0.648·38-s − 1.40·41-s + 0.152·43-s − 0.904·44-s + 0.442·46-s + 1/7·49-s + 0.138·52-s + 1.23·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8696976244$
$L(\frac12)$  $\approx$  $0.8696976244$
$L(\frac{3}{2})$   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 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−8.453884424345087142356650614290, −8.091044136111993741067571884313, −7.38980994272698869605072344429, −6.47517353189684474418126099661, −5.76214448647638297000703306978, −4.97555100712520025613173196251, −3.83517897824153179547742671926, −2.82922405813347082842558391330, −2.09056417572198201018678635137, −0.59867265186005947678509869294, 0.59867265186005947678509869294, 2.09056417572198201018678635137, 2.82922405813347082842558391330, 3.83517897824153179547742671926, 4.97555100712520025613173196251, 5.76214448647638297000703306978, 6.47517353189684474418126099661, 7.38980994272698869605072344429, 8.091044136111993741067571884313, 8.453884424345087142356650614290

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