Properties

Degree $2$
Conductor $250470$
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 + 4-s − 5-s + 8-s − 10-s + 2·13-s + 16-s − 6·17-s − 4·19-s − 20-s + 23-s + 25-s + 2·26-s − 2·29-s + 32-s − 6·34-s − 2·37-s − 4·38-s − 40-s + 10·41-s + 4·43-s + 46-s − 7·49-s + 50-s + 2·52-s − 6·53-s − 2·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s + 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.223·20-s + 0.208·23-s + 1/5·25-s + 0.392·26-s − 0.371·29-s + 0.176·32-s − 1.02·34-s − 0.328·37-s − 0.648·38-s − 0.158·40-s + 1.56·41-s + 0.609·43-s + 0.147·46-s − 49-s + 0.141·50-s + 0.277·52-s − 0.824·53-s − 0.262·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250470\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 23\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{250470} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 250470,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
11 \( 1 \)
23 \( 1 - T \)
good7 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−13.07351584577703, −12.61993712582019, −12.38859129665418, −11.49654786124745, −11.38581644120068, −10.81191348629926, −10.65595756930528, −9.825905404693902, −9.371461190999373, −8.749301815282578, −8.451789519451732, −7.908153292248251, −7.275452098204336, −6.903506465503405, −6.387488955475005, −5.937785314202237, −5.464368168723644, −4.628280145712097, −4.449297056301978, −3.933290249336614, −3.361282808681079, −2.750576759804126, −2.182170700763804, −1.649045526395259, −0.7854820503994483, 0, 0.7854820503994483, 1.649045526395259, 2.182170700763804, 2.750576759804126, 3.361282808681079, 3.933290249336614, 4.449297056301978, 4.628280145712097, 5.464368168723644, 5.937785314202237, 6.387488955475005, 6.903506465503405, 7.275452098204336, 7.908153292248251, 8.451789519451732, 8.749301815282578, 9.371461190999373, 9.825905404693902, 10.65595756930528, 10.81191348629926, 11.38581644120068, 11.49654786124745, 12.38859129665418, 12.61993712582019, 13.07351584577703

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