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 − 2·13-s − 14-s + 16-s + 2·19-s − 2·26-s − 28-s + 6·29-s + 8·31-s + 32-s + 4·37-s + 2·38-s + 6·41-s − 2·43-s + 6·47-s + 49-s − 2·52-s − 6·53-s − 56-s + 6·58-s + 12·59-s + 8·61-s + 8·62-s + 64-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.458·19-s − 0.392·26-s − 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.657·37-s + 0.324·38-s + 0.937·41-s − 0.304·43-s + 0.875·47-s + 1/7·49-s − 0.277·52-s − 0.824·53-s − 0.133·56-s + 0.787·58-s + 1.56·59-s + 1.02·61-s + 1.01·62-s + 1/8·64-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  =  $(2,\ 3150,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.854621188$
$L(\frac12)$  $\approx$  $2.854621188$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−8.583295435803264526433628743831, −7.83484788492072391147649423168, −7.03243190705667838716624797088, −6.38185295352556876478721788069, −5.60655308738568935898973053621, −4.77365969038311726734736492284, −4.07381421480739036927784063240, −3.03657073871473273126945511382, −2.37713756717188893339656403534, −0.932612211263586597747614963660, 0.932612211263586597747614963660, 2.37713756717188893339656403534, 3.03657073871473273126945511382, 4.07381421480739036927784063240, 4.77365969038311726734736492284, 5.60655308738568935898973053621, 6.38185295352556876478721788069, 7.03243190705667838716624797088, 7.83484788492072391147649423168, 8.583295435803264526433628743831

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