Properties

Label 2-4400-1.1-c1-0-45
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  + 2.79·3-s + 0.208·7-s + 4.79·9-s + 11-s − 13-s + 0.791·17-s − 6.58·19-s + 0.582·21-s + 3.79·23-s + 4.99·27-s + 6.79·29-s + 8.58·31-s + 2.79·33-s − 2.58·37-s − 2.79·39-s − 1.41·41-s + 10·43-s + 1.41·47-s − 6.95·49-s + 2.20·51-s + 11.3·53-s − 18.3·57-s + 10.5·59-s + 4.20·61-s + 0.999·63-s + 4·67-s + 10.5·69-s + ⋯
L(s)  = 1  + 1.61·3-s + 0.0788·7-s + 1.59·9-s + 0.301·11-s − 0.277·13-s + 0.191·17-s − 1.51·19-s + 0.127·21-s + 0.790·23-s + 0.962·27-s + 1.26·29-s + 1.54·31-s + 0.485·33-s − 0.424·37-s − 0.446·39-s − 0.221·41-s + 1.52·43-s + 0.206·47-s − 0.993·49-s + 0.309·51-s + 1.56·53-s − 2.43·57-s + 1.37·59-s + 0.538·61-s + 0.125·63-s + 0.488·67-s + 1.27·69-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: $\chi_{4400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.768611506\)
\(L(\frac12)\) \(\approx\) \(3.768611506\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.79T + 3T^{2} \)
7 \( 1 - 0.208T + 7T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 - 0.791T + 17T^{2} \)
19 \( 1 + 6.58T + 19T^{2} \)
23 \( 1 - 3.79T + 23T^{2} \)
29 \( 1 - 6.79T + 29T^{2} \)
31 \( 1 - 8.58T + 31T^{2} \)
37 \( 1 + 2.58T + 37T^{2} \)
41 \( 1 + 1.41T + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 - 1.41T + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 - 4.20T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + 7.79T + 73T^{2} \)
79 \( 1 - 15.5T + 79T^{2} \)
83 \( 1 - 9.95T + 83T^{2} \)
89 \( 1 + 0.791T + 89T^{2} \)
97 \( 1 + 6.20T + 97T^{2} \)
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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

−8.328405098369581229062770170256, −7.959797382753944072572024644275, −6.92465015189509998346135590936, −6.50310068300794013093100224175, −5.25114435780144175260731942785, −4.33727799683710847669317652175, −3.76864890588120928453687305295, −2.72559048907944102315234968023, −2.29438917904943086374888215499, −1.05728421614977377908413900746, 1.05728421614977377908413900746, 2.29438917904943086374888215499, 2.72559048907944102315234968023, 3.76864890588120928453687305295, 4.33727799683710847669317652175, 5.25114435780144175260731942785, 6.50310068300794013093100224175, 6.92465015189509998346135590936, 7.959797382753944072572024644275, 8.328405098369581229062770170256

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