Properties

Label 2-4608-1.1-c1-0-10
Degree $2$
Conductor $4608$
Sign $1$
Analytic cond. $36.7950$
Root an. cond. $6.06589$
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  − 0.449·5-s − 2.04·7-s − 3.46·11-s − 4.79·25-s + 9.34·29-s − 3.60·31-s + 0.921·35-s − 2.79·49-s + 7.55·53-s + 1.55·55-s − 11.3·59-s + 9.79·73-s + 7.10·77-s + 17.4·79-s + 17.3·83-s − 2·97-s + 16.4·101-s + 0.492·103-s − 11.3·107-s + ⋯
L(s)  = 1  − 0.201·5-s − 0.774·7-s − 1.04·11-s − 0.959·25-s + 1.73·29-s − 0.647·31-s + 0.155·35-s − 0.399·49-s + 1.03·53-s + 0.209·55-s − 1.47·59-s + 1.14·73-s + 0.809·77-s + 1.96·79-s + 1.90·83-s − 0.203·97-s + 1.63·101-s + 0.0485·103-s − 1.09·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.171575595\)
\(L(\frac12)\) \(\approx\) \(1.171575595\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + 0.449T + 5T^{2} \)
7 \( 1 + 2.04T + 7T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 9.34T + 29T^{2} \)
31 \( 1 + 3.60T + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 7.55T + 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 9.79T + 73T^{2} \)
79 \( 1 - 17.4T + 79T^{2} \)
83 \( 1 - 17.3T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−8.144633369341423834853447350098, −7.73756911161141834611494618007, −6.80912890772934283225729495968, −6.20656166274815553886594958606, −5.37296693200450229726460147129, −4.63961203850706955314039903208, −3.66949506639917000478347621673, −2.94268475704182839477958847624, −2.05838866376346706104409436290, −0.57710219870707966830547024501, 0.57710219870707966830547024501, 2.05838866376346706104409436290, 2.94268475704182839477958847624, 3.66949506639917000478347621673, 4.63961203850706955314039903208, 5.37296693200450229726460147129, 6.20656166274815553886594958606, 6.80912890772934283225729495968, 7.73756911161141834611494618007, 8.144633369341423834853447350098

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