Properties

Degree $2$
Conductor $287490$
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 − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 3·11-s − 12-s − 3·13-s + 14-s + 15-s + 16-s − 4·17-s − 18-s − 7·19-s − 20-s + 21-s − 3·22-s + 7·23-s + 24-s + 25-s + 3·26-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 + 0.904·11-s − 0.288·12-s − 0.832·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 1.60·19-s − 0.223·20-s + 0.218·21-s − 0.639·22-s + 1.45·23-s + 0.204·24-s + 1/5·25-s + 0.588·26-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: \(1\)
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 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−12.73007523877598, −12.44219806169115, −11.93916123616318, −11.58229272279622, −11.02390760725323, −10.63780733337839, −10.37971575948168, −9.642110738237682, −9.266429477881028, −8.796051563233809, −8.500893654136946, −7.819642891162139, −7.183608667890595, −6.948065154896493, −6.458184530186357, −6.122112049530357, −5.405837389647952, −4.730790360738657, −4.392026271789767, −3.862983492246196, −3.147635328220927, −2.536774087609310, −2.027127054496093, −1.271457094713907, −0.6078788669779889, 0, 0.6078788669779889, 1.271457094713907, 2.027127054496093, 2.536774087609310, 3.147635328220927, 3.862983492246196, 4.392026271789767, 4.730790360738657, 5.405837389647952, 6.122112049530357, 6.458184530186357, 6.948065154896493, 7.183608667890595, 7.819642891162139, 8.500893654136946, 8.796051563233809, 9.266429477881028, 9.642110738237682, 10.37971575948168, 10.63780733337839, 11.02390760725323, 11.58229272279622, 11.93916123616318, 12.44219806169115, 12.73007523877598

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