Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 167 $
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 + 4-s + 5-s + 7-s + 8-s + 10-s + 4·13-s + 14-s + 16-s − 8·17-s − 7·19-s + 20-s − 7·23-s + 25-s + 4·26-s + 28-s + 2·29-s − 2·31-s + 32-s − 8·34-s + 35-s + 12·37-s − 7·38-s + 40-s − 7·41-s + 4·43-s − 7·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 1.94·17-s − 1.60·19-s + 0.223·20-s − 1.45·23-s + 1/5·25-s + 0.784·26-s + 0.188·28-s + 0.371·29-s − 0.359·31-s + 0.176·32-s − 1.37·34-s + 0.169·35-s + 1.97·37-s − 1.13·38-s + 0.158·40-s − 1.09·41-s + 0.609·43-s − 1.03·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(15030\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 167\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{15030} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 15030,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;167\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;167\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 - T \)
167 \( 1 + T \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−16.29636616238667, −15.51357984384146, −15.34355505470839, −14.55655317464185, −14.09283711619484, −13.45903100735002, −13.10215436911557, −12.62322034682189, −11.81783226419721, −11.18300689167133, −10.87746901373587, −10.28151634913050, −9.475879589983391, −8.754807528904671, −8.326230568948228, −7.649061861324512, −6.636074719584479, −6.309974167345956, −5.882818799695293, −4.883931048429763, −4.280673679695642, −3.916023957307884, −2.761773497454001, −2.124873381704515, −1.475468574140264, 0, 1.475468574140264, 2.124873381704515, 2.761773497454001, 3.916023957307884, 4.280673679695642, 4.883931048429763, 5.882818799695293, 6.309974167345956, 6.636074719584479, 7.649061861324512, 8.326230568948228, 8.754807528904671, 9.475879589983391, 10.28151634913050, 10.87746901373587, 11.18300689167133, 11.81783226419721, 12.62322034682189, 13.10215436911557, 13.45903100735002, 14.09283711619484, 14.55655317464185, 15.34355505470839, 15.51357984384146, 16.29636616238667

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