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Properties

Label 2-53550-1.1-c1-0-44
Degree $2$
Conductor $53550$
Sign $-1$
Analytic cond. $427.598$
Root an. cond. $20.6784$
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 + 7-s − 8-s − 4·11-s − 6·13-s − 14-s + 16-s + 17-s + 19-s + 4·22-s + 4·23-s + 6·26-s + 28-s − 3·29-s − 8·31-s − 32-s − 34-s + 6·37-s − 38-s − 2·41-s − 2·43-s − 4·44-s − 4·46-s − 8·47-s + 49-s − 6·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 1.20·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.229·19-s + 0.852·22-s + 0.834·23-s + 1.17·26-s + 0.188·28-s − 0.557·29-s − 1.43·31-s − 0.176·32-s − 0.171·34-s + 0.986·37-s − 0.162·38-s − 0.312·41-s − 0.304·43-s − 0.603·44-s − 0.589·46-s − 1.16·47-s + 1/7·49-s − 0.832·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(53550\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 17\)
Sign: $-1$
Analytic conductor: \(427.598\)
Root analytic conductor: \(20.6784\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 53550,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
17 \( 1 - T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−14.86048567017643, −14.45887215389261, −13.61036875989277, −12.98459629741053, −12.76131954672805, −12.03080059629755, −11.58610093082669, −10.99677437636962, −10.56255936317807, −10.03591965948876, −9.435422557987492, −9.214609572943748, −8.249230771200689, −7.975209772069204, −7.309574430787640, −7.142988821969594, −6.271158838946245, −5.506356610196176, −5.065586888287833, −4.663692022311633, −3.593415918449853, −3.006208802117478, −2.285491918973981, −1.861910304356582, −0.7678915237045756, 0, 0.7678915237045756, 1.861910304356582, 2.285491918973981, 3.006208802117478, 3.593415918449853, 4.663692022311633, 5.065586888287833, 5.506356610196176, 6.271158838946245, 7.142988821969594, 7.309574430787640, 7.975209772069204, 8.249230771200689, 9.214609572943748, 9.435422557987492, 10.03591965948876, 10.56255936317807, 10.99677437636962, 11.58610093082669, 12.03080059629755, 12.76131954672805, 12.98459629741053, 13.61036875989277, 14.45887215389261, 14.86048567017643

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