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Properties

Label 2-53550-1.1-c1-0-100
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 − 2·13-s + 14-s + 16-s + 17-s + 4·19-s − 4·22-s + 6·23-s − 2·26-s + 28-s + 2·29-s − 2·31-s + 32-s + 34-s − 10·37-s + 4·38-s + 6·41-s + 2·43-s − 4·44-s + 6·46-s − 8·47-s + 49-s − 2·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 1.20·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.917·19-s − 0.852·22-s + 1.25·23-s − 0.392·26-s + 0.188·28-s + 0.371·29-s − 0.359·31-s + 0.176·32-s + 0.171·34-s − 1.64·37-s + 0.648·38-s + 0.937·41-s + 0.304·43-s − 0.603·44-s + 0.884·46-s − 1.16·47-s + 1/7·49-s − 0.277·52-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 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
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

−14.54448620615226, −14.28678638103815, −13.63613808518544, −13.15590161181428, −12.77955949292172, −12.14253795765240, −11.78744845291661, −11.09579876005623, −10.71215616385367, −10.16945614588656, −9.660350867631134, −8.896989879326906, −8.442302126006792, −7.531373747588030, −7.486991718934674, −6.846110065510996, −5.999264526415680, −5.495730880074247, −4.944242950303904, −4.684328969699122, −3.754412508788419, −3.046889572586130, −2.708318240478822, −1.847736273466125, −1.096070940243689, 0, 1.096070940243689, 1.847736273466125, 2.708318240478822, 3.046889572586130, 3.754412508788419, 4.684328969699122, 4.944242950303904, 5.495730880074247, 5.999264526415680, 6.846110065510996, 7.486991718934674, 7.531373747588030, 8.442302126006792, 8.896989879326906, 9.660350867631134, 10.16945614588656, 10.71215616385367, 11.09579876005623, 11.78744845291661, 12.14253795765240, 12.77955949292172, 13.15590161181428, 13.63613808518544, 14.28678638103815, 14.54448620615226

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