Properties

Label 2-4410-1.1-c1-0-20
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 + 4·11-s + 2·13-s + 16-s + 2·17-s − 4·19-s + 20-s − 4·22-s + 8·23-s + 25-s − 2·26-s + 2·29-s − 32-s − 2·34-s + 6·37-s + 4·38-s − 40-s − 6·41-s − 4·43-s + 4·44-s − 8·46-s − 50-s + 2·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 1.20·11-s + 0.554·13-s + 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.223·20-s − 0.852·22-s + 1.66·23-s + 1/5·25-s − 0.392·26-s + 0.371·29-s − 0.176·32-s − 0.342·34-s + 0.986·37-s + 0.648·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-s + 0.603·44-s − 1.17·46-s − 0.141·50-s + 0.277·52-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

\(L(1)\) \(\approx\) \(1.791015495\)
\(L(\frac12)\) \(\approx\) \(1.791015495\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.647327749637491427829116510704, −7.68131261207395042033915827358, −6.81537318886956844914397666067, −6.40973998919568114415810052240, −5.59102503422980904290428466069, −4.62242475448465894529250370260, −3.67516322565973252877181398017, −2.79601598717148031870127332799, −1.69559251292467624738091785885, −0.897247117961030373363801846777, 0.897247117961030373363801846777, 1.69559251292467624738091785885, 2.79601598717148031870127332799, 3.67516322565973252877181398017, 4.62242475448465894529250370260, 5.59102503422980904290428466069, 6.40973998919568114415810052240, 6.81537318886956844914397666067, 7.68131261207395042033915827358, 8.647327749637491427829116510704

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