Properties

Label 2-5220-1.1-c1-0-15
Degree $2$
Conductor $5220$
Sign $1$
Analytic cond. $41.6819$
Root an. cond. $6.45615$
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  + 5-s + 2·11-s − 2·13-s − 2·19-s + 8·23-s + 25-s − 29-s + 2·31-s − 4·37-s + 10·41-s + 4·43-s − 12·47-s − 7·49-s + 6·53-s + 2·55-s + 12·59-s − 10·61-s − 2·65-s + 12·67-s − 12·71-s + 12·73-s + 2·79-s + 4·83-s + 10·89-s − 2·95-s + 8·97-s + 6·101-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.603·11-s − 0.554·13-s − 0.458·19-s + 1.66·23-s + 1/5·25-s − 0.185·29-s + 0.359·31-s − 0.657·37-s + 1.56·41-s + 0.609·43-s − 1.75·47-s − 49-s + 0.824·53-s + 0.269·55-s + 1.56·59-s − 1.28·61-s − 0.248·65-s + 1.46·67-s − 1.42·71-s + 1.40·73-s + 0.225·79-s + 0.439·83-s + 1.05·89-s − 0.205·95-s + 0.812·97-s + 0.597·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5220\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 29\)
Sign: $1$
Analytic conductor: \(41.6819\)
Root analytic conductor: \(6.45615\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5220,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.200375115\)
\(L(\frac12)\) \(\approx\) \(2.200375115\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
29 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−8.250401784634935116424626951763, −7.40033147493330085989839512369, −6.74495336694957776906239452941, −6.14396794113848450870733092738, −5.21459022786390552201057580365, −4.65076926295003452163252799768, −3.67639923117339100925767909920, −2.80202439488109963705600728719, −1.90973723859044679527840059360, −0.818507504315089756918734201418, 0.818507504315089756918734201418, 1.90973723859044679527840059360, 2.80202439488109963705600728719, 3.67639923117339100925767909920, 4.65076926295003452163252799768, 5.21459022786390552201057580365, 6.14396794113848450870733092738, 6.74495336694957776906239452941, 7.40033147493330085989839512369, 8.250401784634935116424626951763

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