Properties

Label 2-2200-1.1-c3-0-74
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  − 7·3-s − 2·7-s + 22·9-s − 11·11-s + 38·17-s + 44·19-s + 14·21-s − 175·23-s + 35·27-s − 264·29-s + 159·31-s + 77·33-s + 173·37-s − 220·41-s + 542·43-s + 264·47-s − 339·49-s − 266·51-s − 682·53-s − 308·57-s + 421·59-s + 308·61-s − 44·63-s − 177·67-s + 1.22e3·69-s + 365·71-s + 528·73-s + ⋯
L(s)  = 1  − 1.34·3-s − 0.107·7-s + 0.814·9-s − 0.301·11-s + 0.542·17-s + 0.531·19-s + 0.145·21-s − 1.58·23-s + 0.249·27-s − 1.69·29-s + 0.921·31-s + 0.406·33-s + 0.768·37-s − 0.838·41-s + 1.92·43-s + 0.819·47-s − 0.988·49-s − 0.730·51-s − 1.76·53-s − 0.715·57-s + 0.928·59-s + 0.646·61-s − 0.0879·63-s − 0.322·67-s + 2.13·69-s + 0.610·71-s + 0.846·73-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 \( 1 + 7 T + p^{3} T^{2} \)
7 \( 1 + 2 T + p^{3} T^{2} \)
13 \( 1 + p^{3} T^{2} \)
17 \( 1 - 38 T + p^{3} T^{2} \)
19 \( 1 - 44 T + p^{3} T^{2} \)
23 \( 1 + 175 T + p^{3} T^{2} \)
29 \( 1 + 264 T + p^{3} T^{2} \)
31 \( 1 - 159 T + p^{3} T^{2} \)
37 \( 1 - 173 T + p^{3} T^{2} \)
41 \( 1 + 220 T + p^{3} T^{2} \)
43 \( 1 - 542 T + p^{3} T^{2} \)
47 \( 1 - 264 T + p^{3} T^{2} \)
53 \( 1 + 682 T + p^{3} T^{2} \)
59 \( 1 - 421 T + p^{3} T^{2} \)
61 \( 1 - 308 T + p^{3} T^{2} \)
67 \( 1 + 177 T + p^{3} T^{2} \)
71 \( 1 - 365 T + p^{3} T^{2} \)
73 \( 1 - 528 T + p^{3} T^{2} \)
79 \( 1 - 686 T + p^{3} T^{2} \)
83 \( 1 + 698 T + p^{3} T^{2} \)
89 \( 1 - 967 T + p^{3} T^{2} \)
97 \( 1 - 1127 T + p^{3} T^{2} \)
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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

−8.095161529559578691491816958016, −7.53900469310023824426214709276, −6.52040614208452285065630385898, −5.88996706737237066899080935475, −5.32305886312646487462742220418, −4.42891755604077804377806498637, −3.47153839313739350227777045787, −2.19885896352718543239418690925, −0.956110229762857378118687793070, 0, 0.956110229762857378118687793070, 2.19885896352718543239418690925, 3.47153839313739350227777045787, 4.42891755604077804377806498637, 5.32305886312646487462742220418, 5.88996706737237066899080935475, 6.52040614208452285065630385898, 7.53900469310023824426214709276, 8.095161529559578691491816958016

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