Properties

Label 2-4410-1.1-c1-0-10
Degree $2$
Conductor $4410$
Sign $1$
Analytic cond. $35.2140$
Root an. cond. $5.93414$
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-s + 4-s − 5-s + 8-s − 10-s − 3·11-s − 5·13-s + 16-s − 5·19-s − 20-s − 3·22-s + 9·23-s + 25-s − 5·26-s + 10·31-s + 32-s − 37-s − 5·38-s − 40-s + 9·41-s + 8·43-s − 3·44-s + 9·46-s + 3·47-s + 50-s − 5·52-s + 3·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 0.904·11-s − 1.38·13-s + 1/4·16-s − 1.14·19-s − 0.223·20-s − 0.639·22-s + 1.87·23-s + 1/5·25-s − 0.980·26-s + 1.79·31-s + 0.176·32-s − 0.164·37-s − 0.811·38-s − 0.158·40-s + 1.40·41-s + 1.21·43-s − 0.452·44-s + 1.32·46-s + 0.437·47-s + 0.141·50-s − 0.693·52-s + 0.412·53-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.367396791\)
\(L(\frac12)\) \(\approx\) \(2.367396791\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−8.191418000804445902345731363397, −7.49949362666047166694334974988, −6.94732894842194039744888170272, −6.13046960376428895622096169324, −5.15687452530896148305352390636, −4.72195433547146702776735748504, −3.93542760933010284146035376443, −2.74612032996774865758309893257, −2.42925746146616335834036074022, −0.75382160053780565454019409712, 0.75382160053780565454019409712, 2.42925746146616335834036074022, 2.74612032996774865758309893257, 3.93542760933010284146035376443, 4.72195433547146702776735748504, 5.15687452530896148305352390636, 6.13046960376428895622096169324, 6.94732894842194039744888170272, 7.49949362666047166694334974988, 8.191418000804445902345731363397

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