Properties

Degree $2$
Conductor $50430$
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 + 4·7-s − 8-s + 9-s + 10-s − 12-s − 2·13-s − 4·14-s + 15-s + 16-s − 6·17-s − 18-s + 4·19-s − 20-s − 4·21-s + 24-s + 25-s + 2·26-s − 27-s + 4·28-s + 6·29-s − 30-s + 8·31-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.554·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.872·21-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.755·28-s + 1.11·29-s − 0.182·30-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50430\)    =    \(2 \cdot 3 \cdot 5 \cdot 41^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{50430} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 50430,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
41 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−14.98139614175195, −14.32796917770696, −13.71652012534002, −13.36033643531488, −12.32539360860126, −12.09788450036500, −11.57450012928400, −11.17212427469370, −10.76214311062696, −10.17009658044904, −9.629417047601413, −8.953699368183665, −8.360607591417932, −8.044233999634391, −7.459730931170358, −6.855321544899906, −6.455969333388876, −5.585576633390205, −4.995811958718539, −4.546138467950438, −4.044613355052554, −2.907177173959177, −2.383142441438017, −1.511183778078476, −0.9511257119263556, 0, 0.9511257119263556, 1.511183778078476, 2.383142441438017, 2.907177173959177, 4.044613355052554, 4.546138467950438, 4.995811958718539, 5.585576633390205, 6.455969333388876, 6.855321544899906, 7.459730931170358, 8.044233999634391, 8.360607591417932, 8.953699368183665, 9.629417047601413, 10.17009658044904, 10.76214311062696, 11.17212427469370, 11.57450012928400, 12.09788450036500, 12.32539360860126, 13.36033643531488, 13.71652012534002, 14.32796917770696, 14.98139614175195

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