Properties

Label 2-2200-1.1-c3-0-122
Degree $2$
Conductor $2200$
Sign $-1$
Analytic cond. $129.804$
Root an. cond. $11.3931$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·3-s − 7-s − 2·9-s + 11·11-s − 18·13-s + 113·17-s + 55·19-s − 5·21-s − 190·23-s − 145·27-s − 69·29-s − 255·31-s + 55·33-s − 51·37-s − 90·39-s − 314·41-s + 484·43-s − 470·47-s − 342·49-s + 565·51-s + 545·53-s + 275·57-s − 102·59-s + 129·61-s + 2·63-s + 664·67-s − 950·69-s + ⋯
L(s)  = 1  + 0.962·3-s − 0.0539·7-s − 0.0740·9-s + 0.301·11-s − 0.384·13-s + 1.61·17-s + 0.664·19-s − 0.0519·21-s − 1.72·23-s − 1.03·27-s − 0.441·29-s − 1.47·31-s + 0.290·33-s − 0.226·37-s − 0.369·39-s − 1.19·41-s + 1.71·43-s − 1.45·47-s − 0.997·49-s + 1.55·51-s + 1.41·53-s + 0.639·57-s − 0.225·59-s + 0.270·61-s + 0.00399·63-s + 1.21·67-s − 1.65·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(129.804\)
Root analytic conductor: \(11.3931\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2200,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 - p T \)
good3 \( 1 - 5 T + p^{3} T^{2} \)
7 \( 1 + T + p^{3} T^{2} \)
13 \( 1 + 18 T + p^{3} T^{2} \)
17 \( 1 - 113 T + p^{3} T^{2} \)
19 \( 1 - 55 T + p^{3} T^{2} \)
23 \( 1 + 190 T + p^{3} T^{2} \)
29 \( 1 + 69 T + p^{3} T^{2} \)
31 \( 1 + 255 T + p^{3} T^{2} \)
37 \( 1 + 51 T + p^{3} T^{2} \)
41 \( 1 + 314 T + p^{3} T^{2} \)
43 \( 1 - 484 T + p^{3} T^{2} \)
47 \( 1 + 10 p T + p^{3} T^{2} \)
53 \( 1 - 545 T + p^{3} T^{2} \)
59 \( 1 + 102 T + p^{3} T^{2} \)
61 \( 1 - 129 T + p^{3} T^{2} \)
67 \( 1 - 664 T + p^{3} T^{2} \)
71 \( 1 + 1029 T + p^{3} T^{2} \)
73 \( 1 - 758 T + p^{3} T^{2} \)
79 \( 1 - 634 T + p^{3} T^{2} \)
83 \( 1 - 654 T + p^{3} T^{2} \)
89 \( 1 + 511 T + p^{3} T^{2} \)
97 \( 1 + 1736 T + p^{3} T^{2} \)
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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

−8.122882659018850403709173533087, −7.82733996292729670605214217234, −6.94644486564563250367390675315, −5.82158558668888585553556932394, −5.25427114651971223593293101240, −3.87770569861044140617593200753, −3.41599871609523956783466301442, −2.39303820824050369049410384351, −1.45878566223800258691030729135, 0, 1.45878566223800258691030729135, 2.39303820824050369049410384351, 3.41599871609523956783466301442, 3.87770569861044140617593200753, 5.25427114651971223593293101240, 5.82158558668888585553556932394, 6.94644486564563250367390675315, 7.82733996292729670605214217234, 8.122882659018850403709173533087

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