Properties

Label 2-4400-1.1-c1-0-59
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4·7-s + 9-s + 11-s + 5·13-s + 7·19-s + 8·21-s − 3·23-s − 4·27-s + 3·29-s − 5·31-s + 2·33-s − 4·37-s + 10·39-s + 12·41-s − 5·43-s + 9·49-s + 6·53-s + 14·57-s − 12·59-s − 10·61-s + 4·63-s − 14·67-s − 6·69-s − 3·71-s + 8·73-s + 4·77-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.51·7-s + 1/3·9-s + 0.301·11-s + 1.38·13-s + 1.60·19-s + 1.74·21-s − 0.625·23-s − 0.769·27-s + 0.557·29-s − 0.898·31-s + 0.348·33-s − 0.657·37-s + 1.60·39-s + 1.87·41-s − 0.762·43-s + 9/7·49-s + 0.824·53-s + 1.85·57-s − 1.56·59-s − 1.28·61-s + 0.503·63-s − 1.71·67-s − 0.722·69-s − 0.356·71-s + 0.936·73-s + 0.455·77-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: $\chi_{4400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.988831941\)
\(L(\frac12)\) \(\approx\) \(3.988831941\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−8.215066690317924879429209850018, −7.897985588647675963820965641286, −7.23198558052927413132140922600, −6.07659462756156825046100662425, −5.40373239996389746725721759481, −4.46571875991929313351266021179, −3.69956159581703457314506928919, −2.97493428037670400673680749715, −1.87861636891240765312856224651, −1.21298979508838572831757993102, 1.21298979508838572831757993102, 1.87861636891240765312856224651, 2.97493428037670400673680749715, 3.69956159581703457314506928919, 4.46571875991929313351266021179, 5.40373239996389746725721759481, 6.07659462756156825046100662425, 7.23198558052927413132140922600, 7.897985588647675963820965641286, 8.215066690317924879429209850018

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