Properties

Label 2-2200-1.1-c3-0-42
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 6·7-s − 26·9-s − 11·11-s + 40·13-s + 78·17-s + 36·19-s + 6·21-s − 7·23-s − 53·27-s + 8·29-s + 183·31-s − 11·33-s − 227·37-s + 40·39-s − 36·41-s − 322·43-s + 184·47-s − 307·49-s + 78·51-s + 6·53-s + 36·57-s − 99·59-s + 164·61-s − 156·63-s + 695·67-s − 7·69-s + ⋯
L(s)  = 1  + 0.192·3-s + 0.323·7-s − 0.962·9-s − 0.301·11-s + 0.853·13-s + 1.11·17-s + 0.434·19-s + 0.0623·21-s − 0.0634·23-s − 0.377·27-s + 0.0512·29-s + 1.06·31-s − 0.0580·33-s − 1.00·37-s + 0.164·39-s − 0.137·41-s − 1.14·43-s + 0.571·47-s − 0.895·49-s + 0.214·51-s + 0.0155·53-s + 0.0836·57-s − 0.218·59-s + 0.344·61-s − 0.311·63-s + 1.26·67-s − 0.0122·69-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: \(0\)
Selberg data: \((2,\ 2200,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.309266238\)
\(L(\frac12)\) \(\approx\) \(2.309266238\)
\(L(\frac{5}{2})\) 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 \( 1 - T + p^{3} T^{2} \)
7 \( 1 - 6 T + p^{3} T^{2} \)
13 \( 1 - 40 T + p^{3} T^{2} \)
17 \( 1 - 78 T + p^{3} T^{2} \)
19 \( 1 - 36 T + p^{3} T^{2} \)
23 \( 1 + 7 T + p^{3} T^{2} \)
29 \( 1 - 8 T + p^{3} T^{2} \)
31 \( 1 - 183 T + p^{3} T^{2} \)
37 \( 1 + 227 T + p^{3} T^{2} \)
41 \( 1 + 36 T + p^{3} T^{2} \)
43 \( 1 + 322 T + p^{3} T^{2} \)
47 \( 1 - 184 T + p^{3} T^{2} \)
53 \( 1 - 6 T + p^{3} T^{2} \)
59 \( 1 + 99 T + p^{3} T^{2} \)
61 \( 1 - 164 T + p^{3} T^{2} \)
67 \( 1 - 695 T + p^{3} T^{2} \)
71 \( 1 + 987 T + p^{3} T^{2} \)
73 \( 1 - 248 T + p^{3} T^{2} \)
79 \( 1 + 242 T + p^{3} T^{2} \)
83 \( 1 - 18 p T + p^{3} T^{2} \)
89 \( 1 + 905 T + p^{3} T^{2} \)
97 \( 1 - 1031 T + p^{3} T^{2} \)
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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

−8.448559384931573013498080047562, −8.189483653280291706648919485513, −7.24302818106195842803260565351, −6.25079540145578632348310728102, −5.55266099954697860685089591505, −4.81029287707479781757483372455, −3.58073962071023748266228978086, −2.99074027308913228947860352853, −1.79829361275960864062231864103, −0.68654674140118427980815271000, 0.68654674140118427980815271000, 1.79829361275960864062231864103, 2.99074027308913228947860352853, 3.58073962071023748266228978086, 4.81029287707479781757483372455, 5.55266099954697860685089591505, 6.25079540145578632348310728102, 7.24302818106195842803260565351, 8.189483653280291706648919485513, 8.448559384931573013498080047562

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