Properties

Label 2-2200-1.1-c3-0-113
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  − 6·3-s + 32·7-s + 9·9-s + 11·11-s + 48·13-s + 36·17-s − 44·19-s − 192·21-s − 58·23-s + 108·27-s − 278·29-s − 112·31-s − 66·33-s − 194·37-s − 288·39-s − 314·41-s − 396·43-s + 410·47-s + 681·49-s − 216·51-s − 170·53-s + 264·57-s + 404·59-s + 250·61-s + 288·63-s + 26·67-s + 348·69-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.72·7-s + 1/3·9-s + 0.301·11-s + 1.02·13-s + 0.513·17-s − 0.531·19-s − 1.99·21-s − 0.525·23-s + 0.769·27-s − 1.78·29-s − 0.648·31-s − 0.348·33-s − 0.861·37-s − 1.18·39-s − 1.19·41-s − 1.40·43-s + 1.27·47-s + 1.98·49-s − 0.593·51-s − 0.440·53-s + 0.613·57-s + 0.891·59-s + 0.524·61-s + 0.575·63-s + 0.0474·67-s + 0.607·69-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 \( 1 + 2 p T + p^{3} T^{2} \)
7 \( 1 - 32 T + p^{3} T^{2} \)
13 \( 1 - 48 T + p^{3} T^{2} \)
17 \( 1 - 36 T + p^{3} T^{2} \)
19 \( 1 + 44 T + p^{3} T^{2} \)
23 \( 1 + 58 T + p^{3} T^{2} \)
29 \( 1 + 278 T + p^{3} T^{2} \)
31 \( 1 + 112 T + p^{3} T^{2} \)
37 \( 1 + 194 T + p^{3} T^{2} \)
41 \( 1 + 314 T + p^{3} T^{2} \)
43 \( 1 + 396 T + p^{3} T^{2} \)
47 \( 1 - 410 T + p^{3} T^{2} \)
53 \( 1 + 170 T + p^{3} T^{2} \)
59 \( 1 - 404 T + p^{3} T^{2} \)
61 \( 1 - 250 T + p^{3} T^{2} \)
67 \( 1 - 26 T + p^{3} T^{2} \)
71 \( 1 + 468 T + p^{3} T^{2} \)
73 \( 1 - 164 T + p^{3} T^{2} \)
79 \( 1 + 664 T + p^{3} T^{2} \)
83 \( 1 + 1348 T + p^{3} T^{2} \)
89 \( 1 - 6 p T + p^{3} T^{2} \)
97 \( 1 - 1498 T + p^{3} T^{2} \)
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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

−8.390320408434897671180724627975, −7.50665863923987442990705320585, −6.67628675049067213703977471912, −5.66299864413597000390671409708, −5.37156751506603335113616236056, −4.41345426982767282175366810864, −3.58319864893154670308428396387, −1.93138179365006712460492437224, −1.26370169868365972221051445356, 0, 1.26370169868365972221051445356, 1.93138179365006712460492437224, 3.58319864893154670308428396387, 4.41345426982767282175366810864, 5.37156751506603335113616236056, 5.66299864413597000390671409708, 6.67628675049067213703977471912, 7.50665863923987442990705320585, 8.390320408434897671180724627975

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