Properties

Degree $2$
Conductor $287490$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $2$

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 − 4·11-s + 12-s + 2·13-s + 14-s − 15-s + 16-s − 2·17-s − 18-s − 4·19-s − 20-s − 21-s + 4·22-s + 8·23-s − 24-s + 25-s − 2·26-s + 27-s − 28-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 − 1.20·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.218·21-s + 0.852·22-s + 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.188·28-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: \(2\)
Selberg data: \((2,\ 287490,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.15665985605770, −12.65602417226229, −12.45484535346406, −11.71209940271277, −11.06209296497161, −10.94400706333372, −10.34436108561798, −10.09872826886554, −9.316623155223501, −8.878129293496139, −8.749819207454602, −8.059567461758888, −7.710923419571995, −7.267122879362523, −6.735261913663520, −6.260141361457757, −5.773040632859250, −4.934225697295593, −4.643540793682552, −3.975188686092242, −3.277428597651877, −2.816290771488825, −2.579652972806318, −1.582947039819907, −1.242040132653579, 0, 0, 1.242040132653579, 1.582947039819907, 2.579652972806318, 2.816290771488825, 3.277428597651877, 3.975188686092242, 4.643540793682552, 4.934225697295593, 5.773040632859250, 6.260141361457757, 6.735261913663520, 7.267122879362523, 7.710923419571995, 8.059567461758888, 8.749819207454602, 8.878129293496139, 9.316623155223501, 10.09872826886554, 10.34436108561798, 10.94400706333372, 11.06209296497161, 11.71209940271277, 12.45484535346406, 12.65602417226229, 13.15665985605770

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