Properties

Label 2-4400-1.1-c1-0-25
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.618·3-s + 0.381·7-s − 2.61·9-s − 11-s + 5.47·13-s − 5.09·17-s + 3.76·19-s + 0.236·21-s − 0.381·23-s − 3.47·27-s + 8.32·29-s + 8.70·31-s − 0.618·33-s + 0.236·37-s + 3.38·39-s − 10.7·41-s − 6.94·43-s + 2.23·47-s − 6.85·49-s − 3.14·51-s + 3.61·53-s + 2.32·57-s + 10.7·59-s − 1.90·61-s − 63-s + 12·67-s − 0.236·69-s + ⋯
L(s)  = 1  + 0.356·3-s + 0.144·7-s − 0.872·9-s − 0.301·11-s + 1.51·13-s − 1.23·17-s + 0.863·19-s + 0.0515·21-s − 0.0796·23-s − 0.668·27-s + 1.54·29-s + 1.56·31-s − 0.107·33-s + 0.0388·37-s + 0.541·39-s − 1.67·41-s − 1.05·43-s + 0.326·47-s − 0.979·49-s − 0.440·51-s + 0.496·53-s + 0.308·57-s + 1.39·59-s − 0.244·61-s − 0.125·63-s + 1.46·67-s − 0.0284·69-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

\(L(1)\) \(\approx\) \(2.098542295\)
\(L(\frac12)\) \(\approx\) \(2.098542295\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.618T + 3T^{2} \)
7 \( 1 - 0.381T + 7T^{2} \)
13 \( 1 - 5.47T + 13T^{2} \)
17 \( 1 + 5.09T + 17T^{2} \)
19 \( 1 - 3.76T + 19T^{2} \)
23 \( 1 + 0.381T + 23T^{2} \)
29 \( 1 - 8.32T + 29T^{2} \)
31 \( 1 - 8.70T + 31T^{2} \)
37 \( 1 - 0.236T + 37T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 + 6.94T + 43T^{2} \)
47 \( 1 - 2.23T + 47T^{2} \)
53 \( 1 - 3.61T + 53T^{2} \)
59 \( 1 - 10.7T + 59T^{2} \)
61 \( 1 + 1.90T + 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + 8.18T + 71T^{2} \)
73 \( 1 - 11.6T + 73T^{2} \)
79 \( 1 - 7.38T + 79T^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.549086188306114807515407536749, −7.897469879952028997574627456276, −6.68809024919949034091786092554, −6.35840694464300036870193010589, −5.36012444384650687012821119510, −4.67132293343261082011012416090, −3.63079319877428014649182622740, −2.97980383295172907167557654382, −2.03583104278383959820010443173, −0.802396854074297544795514332761, 0.802396854074297544795514332761, 2.03583104278383959820010443173, 2.97980383295172907167557654382, 3.63079319877428014649182622740, 4.67132293343261082011012416090, 5.36012444384650687012821119510, 6.35840694464300036870193010589, 6.68809024919949034091786092554, 7.897469879952028997574627456276, 8.549086188306114807515407536749

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