Properties

Degree $2$
Conductor $287490$
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 − 6·11-s − 12-s − 6·13-s + 14-s − 15-s + 16-s − 6·17-s − 18-s + 20-s + 21-s + 6·22-s + 2·23-s + 24-s + 25-s + 6·26-s − 27-s − 28-s + 3·29-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 − 1.80·11-s − 0.288·12-s − 1.66·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.223·20-s + 0.218·21-s + 1.27·22-s + 0.417·23-s + 0.204·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s − 0.188·28-s + 0.557·29-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

Degree: \(2\)
Conductor: \(287490\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 37^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{287490} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 287490,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3667010358\)
\(L(\frac12)\) \(\approx\) \(0.3667010358\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 + T \)
37 \( 1 \)
good11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.69190397269247, −12.24949484048541, −11.84605125833106, −11.06043833948638, −10.88251460542211, −10.43353335885175, −9.954040217944873, −9.544215100808813, −9.281949129420605, −8.441435243664227, −8.140391240968390, −7.575355330677884, −7.071945703904051, −6.733916338536909, −6.225990589397013, −5.563096697326771, −5.207169063841329, −4.651113883983272, −4.331726339666228, −3.199090811891245, −2.762327797464269, −2.304125928017204, −1.894526826954893, −0.8529683414147491, −0.2293954211997034, 0.2293954211997034, 0.8529683414147491, 1.894526826954893, 2.304125928017204, 2.762327797464269, 3.199090811891245, 4.331726339666228, 4.651113883983272, 5.207169063841329, 5.563096697326771, 6.225990589397013, 6.733916338536909, 7.071945703904051, 7.575355330677884, 8.140391240968390, 8.441435243664227, 9.281949129420605, 9.544215100808813, 9.954040217944873, 10.43353335885175, 10.88251460542211, 11.06043833948638, 11.84605125833106, 12.24949484048541, 12.69190397269247

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