Properties

Label 2-20286-1.1-c1-0-59
Degree $2$
Conductor $20286$
Sign $-1$
Analytic cond. $161.984$
Root an. cond. $12.7273$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 4·11-s + 16-s + 6·17-s + 6·19-s − 4·22-s + 23-s − 5·25-s − 10·29-s − 4·31-s + 32-s + 6·34-s − 2·37-s + 6·38-s − 10·41-s − 4·43-s − 4·44-s + 46-s + 12·47-s − 5·50-s + 6·53-s − 10·58-s − 2·59-s − 4·62-s + 64-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.20·11-s + 1/4·16-s + 1.45·17-s + 1.37·19-s − 0.852·22-s + 0.208·23-s − 25-s − 1.85·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s − 0.328·37-s + 0.973·38-s − 1.56·41-s − 0.609·43-s − 0.603·44-s + 0.147·46-s + 1.75·47-s − 0.707·50-s + 0.824·53-s − 1.31·58-s − 0.260·59-s − 0.508·62-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20286 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20286 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20286\)    =    \(2 \cdot 3^{2} \cdot 7^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(161.984\)
Root analytic conductor: \(12.7273\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{20286} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 20286,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
23 \( 1 - T \)
good5 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−15.74880468743092, −15.36185625683440, −14.87332497936584, −14.17420672084479, −13.69800056616376, −13.30606708920630, −12.63935783929137, −12.17087644786345, −11.59758961275116, −11.08516144769765, −10.38741021716937, −9.890447809945803, −9.398219672823070, −8.483824894582338, −7.809001521985447, −7.411404870380380, −6.928653911408617, −5.774139930171431, −5.466614805585699, −5.184759280082271, −4.071556876892117, −3.504581471851294, −2.940137710062613, −2.075926414980710, −1.264964318251639, 0, 1.264964318251639, 2.075926414980710, 2.940137710062613, 3.504581471851294, 4.071556876892117, 5.184759280082271, 5.466614805585699, 5.774139930171431, 6.928653911408617, 7.411404870380380, 7.809001521985447, 8.483824894582338, 9.398219672823070, 9.890447809945803, 10.38741021716937, 11.08516144769765, 11.59758961275116, 12.17087644786345, 12.63935783929137, 13.30606708920630, 13.69800056616376, 14.17420672084479, 14.87332497936584, 15.36185625683440, 15.74880468743092

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