Properties

Label 2-4400-1.1-c1-0-15
Degree $2$
Conductor $4400$
Sign $1$
Analytic cond. $35.1341$
Root an. cond. $5.92740$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.67·3-s + 4.38·7-s + 4.16·9-s − 11-s − 4·13-s − 5.87·17-s + 0.526·19-s − 11.7·21-s + 6.15·23-s − 3.11·27-s − 0.967·29-s + 9.60·31-s + 2.67·33-s − 1.76·37-s + 10.7·39-s + 2.50·41-s − 3.85·43-s − 4.91·47-s + 12.2·49-s + 15.7·51-s + 5.29·53-s − 1.40·57-s + 2.63·59-s + 8.96·61-s + 18.2·63-s − 7.97·67-s − 16.4·69-s + ⋯
L(s)  = 1  − 1.54·3-s + 1.65·7-s + 1.38·9-s − 0.301·11-s − 1.10·13-s − 1.42·17-s + 0.120·19-s − 2.56·21-s + 1.28·23-s − 0.600·27-s − 0.179·29-s + 1.72·31-s + 0.465·33-s − 0.290·37-s + 1.71·39-s + 0.391·41-s − 0.588·43-s − 0.716·47-s + 1.74·49-s + 2.20·51-s + 0.727·53-s − 0.186·57-s + 0.343·59-s + 1.14·61-s + 2.30·63-s − 0.974·67-s − 1.98·69-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.088350455\)
\(L(\frac12)\) \(\approx\) \(1.088350455\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.67T + 3T^{2} \)
7 \( 1 - 4.38T + 7T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + 5.87T + 17T^{2} \)
19 \( 1 - 0.526T + 19T^{2} \)
23 \( 1 - 6.15T + 23T^{2} \)
29 \( 1 + 0.967T + 29T^{2} \)
31 \( 1 - 9.60T + 31T^{2} \)
37 \( 1 + 1.76T + 37T^{2} \)
41 \( 1 - 2.50T + 41T^{2} \)
43 \( 1 + 3.85T + 43T^{2} \)
47 \( 1 + 4.91T + 47T^{2} \)
53 \( 1 - 5.29T + 53T^{2} \)
59 \( 1 - 2.63T + 59T^{2} \)
61 \( 1 - 8.96T + 61T^{2} \)
67 \( 1 + 7.97T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 - 13.2T + 79T^{2} \)
83 \( 1 + 7.61T + 83T^{2} \)
89 \( 1 - 2.83T + 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−8.327423085750688903686374542036, −7.41922985145783571524892685285, −6.92261331548167742884524612649, −6.08374631032343967430015045234, −5.20210440576040827849996517031, −4.79473987699874021404978148233, −4.36761439253146100401636219166, −2.71962235475991912797017149172, −1.73005073445970647951152534131, −0.64729866599823669313919931637, 0.64729866599823669313919931637, 1.73005073445970647951152534131, 2.71962235475991912797017149172, 4.36761439253146100401636219166, 4.79473987699874021404978148233, 5.20210440576040827849996517031, 6.08374631032343967430015045234, 6.92261331548167742884524612649, 7.41922985145783571524892685285, 8.327423085750688903686374542036

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