Properties

Degree $2$
Conductor $60690$
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 − 4·11-s + 12-s − 2·13-s − 14-s − 15-s + 16-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 − 6·29-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 − 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-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 − 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60690\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 17^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{60690} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 60690,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
17 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−14.70388064888054, −13.99475162062793, −13.39942650599118, −13.04209506254163, −12.78551440059055, −12.19415400976515, −11.42476270792992, −11.21792690501712, −10.36241603668638, −10.12685185118258, −9.484759128546266, −8.754695039593885, −8.272338261992373, −7.814436688670407, −7.176461978804029, −6.752943060812365, −6.200815467560481, −5.286286722052768, −4.938491148685191, −4.450881091197931, −3.507423150885516, −3.294266127053605, −2.463609278782110, −2.123216792542381, −0.9744194568654826, 0, 0.9744194568654826, 2.123216792542381, 2.463609278782110, 3.294266127053605, 3.507423150885516, 4.450881091197931, 4.938491148685191, 5.286286722052768, 6.200815467560481, 6.752943060812365, 7.176461978804029, 7.814436688670407, 8.272338261992373, 8.754695039593885, 9.484759128546266, 10.12685185118258, 10.36241603668638, 11.21792690501712, 11.42476270792992, 12.19415400976515, 12.78551440059055, 13.04209506254163, 13.39942650599118, 13.99475162062793, 14.70388064888054

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