Properties

Label 2-4400-1.1-c1-0-52
Degree $2$
Conductor $4400$
Sign $-1$
Analytic cond. $35.1341$
Root an. cond. $5.92740$
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  − 2·3-s + 9-s − 11-s + 3·13-s − 4·17-s + 19-s − 3·23-s + 4·27-s + 5·29-s + 3·31-s + 2·33-s − 12·37-s − 6·39-s + 8·41-s + 5·43-s + 8·47-s − 7·49-s + 8·51-s − 10·53-s − 2·57-s − 8·59-s + 10·61-s − 14·67-s + 6·69-s + 5·71-s − 4·73-s + 8·79-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 0.301·11-s + 0.832·13-s − 0.970·17-s + 0.229·19-s − 0.625·23-s + 0.769·27-s + 0.928·29-s + 0.538·31-s + 0.348·33-s − 1.97·37-s − 0.960·39-s + 1.24·41-s + 0.762·43-s + 1.16·47-s − 49-s + 1.12·51-s − 1.37·53-s − 0.264·57-s − 1.04·59-s + 1.28·61-s − 1.71·67-s + 0.722·69-s + 0.593·71-s − 0.468·73-s + 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4400\)    =    \(2^{4} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(35.1341\)
Root analytic conductor: \(5.92740\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
11 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−8.009396201086558330755980077068, −7.09661170954696028589073275698, −6.31774208635278566143098758450, −5.95251555269509447771927127170, −5.04137137530561945884073121971, −4.44889853685944012773215613315, −3.44307601477960793257883007200, −2.37942017148615794672168448724, −1.15845751029152603421619719945, 0, 1.15845751029152603421619719945, 2.37942017148615794672168448724, 3.44307601477960793257883007200, 4.44889853685944012773215613315, 5.04137137530561945884073121971, 5.95251555269509447771927127170, 6.31774208635278566143098758450, 7.09661170954696028589073275698, 8.009396201086558330755980077068

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