Properties

Label 2-4410-1.1-c1-0-2
Degree $2$
Conductor $4410$
Sign $1$
Analytic cond. $35.2140$
Root an. cond. $5.93414$
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-s + 4-s − 5-s − 8-s + 10-s − 0.585·11-s + 16-s − 1.41·17-s − 2.82·19-s − 20-s + 0.585·22-s − 4.82·23-s + 25-s − 8.24·29-s + 5.07·31-s − 32-s + 1.41·34-s − 1.41·37-s + 2.82·38-s + 40-s + 8.82·41-s + 4.58·43-s − 0.585·44-s + 4.82·46-s + 9.07·47-s − 50-s + 9.31·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.176·11-s + 0.250·16-s − 0.342·17-s − 0.648·19-s − 0.223·20-s + 0.124·22-s − 1.00·23-s + 0.200·25-s − 1.53·29-s + 0.910·31-s − 0.176·32-s + 0.242·34-s − 0.232·37-s + 0.458·38-s + 0.158·40-s + 1.37·41-s + 0.699·43-s − 0.0883·44-s + 0.711·46-s + 1.32·47-s − 0.141·50-s + 1.27·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(35.2140\)
Root analytic conductor: \(5.93414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9088887485\)
\(L(\frac12)\) \(\approx\) \(0.9088887485\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + 0.585T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 1.41T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 + 4.82T + 23T^{2} \)
29 \( 1 + 8.24T + 29T^{2} \)
31 \( 1 - 5.07T + 31T^{2} \)
37 \( 1 + 1.41T + 37T^{2} \)
41 \( 1 - 8.82T + 41T^{2} \)
43 \( 1 - 4.58T + 43T^{2} \)
47 \( 1 - 9.07T + 47T^{2} \)
53 \( 1 - 9.31T + 53T^{2} \)
59 \( 1 + 2.48T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 - 7.89T + 67T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 - 7.65T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 - 5.17T + 83T^{2} \)
89 \( 1 - 6.48T + 89T^{2} \)
97 \( 1 - 8.82T + 97T^{2} \)
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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

−8.351887475032721664037497919489, −7.66976992285672531886610698417, −7.15991630721081878821854163042, −6.20244475736694238574184177443, −5.64115580524938951581261519207, −4.45805013245717675182016087525, −3.83611390297570794866730714467, −2.71113509120715599062087712455, −1.90324895442588401192918624825, −0.57957080469630182225994034023, 0.57957080469630182225994034023, 1.90324895442588401192918624825, 2.71113509120715599062087712455, 3.83611390297570794866730714467, 4.45805013245717675182016087525, 5.64115580524938951581261519207, 6.20244475736694238574184177443, 7.15991630721081878821854163042, 7.66976992285672531886610698417, 8.351887475032721664037497919489

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