Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 \cdot 37^{2} $
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 − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 12-s − 2·13-s + 14-s − 15-s + 16-s + 6·17-s + 18-s − 8·19-s − 20-s + 21-s + 24-s + 25-s − 2·26-s + 27-s + 28-s − 6·29-s − 30-s + 4·31-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 0.554·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 1.83·19-s − 0.223·20-s + 0.218·21-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s − 0.182·30-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(287490\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 37^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{287490} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 287490,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(4.642613228\)
\(L(\frac12)\)  \(\approx\)  \(4.642613228\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;37\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;37\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
37 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−12.77201435452826, −12.32373440862710, −11.90792025004546, −11.47928118519555, −10.93505761249230, −10.45877286907502, −10.03835433953083, −9.591867937190017, −8.920970817859511, −8.398570704764232, −8.106890551610592, −7.532387395856807, −7.220391507145282, −6.574449010370056, −6.147784231372665, −5.474469836946587, −5.044808933390375, −4.537995436884652, −3.933608179180757, −3.663733860477211, −3.018364057073324, −2.392854886208663, −1.985952722461896, −1.284376806852867, −0.4790899118917553, 0.4790899118917553, 1.284376806852867, 1.985952722461896, 2.392854886208663, 3.018364057073324, 3.663733860477211, 3.933608179180757, 4.537995436884652, 5.044808933390375, 5.474469836946587, 6.147784231372665, 6.574449010370056, 7.220391507145282, 7.532387395856807, 8.106890551610592, 8.398570704764232, 8.920970817859511, 9.591867937190017, 10.03835433953083, 10.45877286907502, 10.93505761249230, 11.47928118519555, 11.90792025004546, 12.32373440862710, 12.77201435452826

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