Properties

Label 2-4598-1.1-c1-0-108
Degree $2$
Conductor $4598$
Sign $1$
Analytic cond. $36.7152$
Root an. cond. $6.05930$
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 + 3·3-s + 4-s + 3·6-s + 8-s + 6·9-s + 3·12-s + 7·13-s + 16-s − 7·17-s + 6·18-s + 19-s + 8·23-s + 3·24-s − 5·25-s + 7·26-s + 9·27-s − 9·29-s + 2·31-s + 32-s − 7·34-s + 6·36-s + 3·37-s + 38-s + 21·39-s + 10·41-s − 10·43-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 1/2·4-s + 1.22·6-s + 0.353·8-s + 2·9-s + 0.866·12-s + 1.94·13-s + 1/4·16-s − 1.69·17-s + 1.41·18-s + 0.229·19-s + 1.66·23-s + 0.612·24-s − 25-s + 1.37·26-s + 1.73·27-s − 1.67·29-s + 0.359·31-s + 0.176·32-s − 1.20·34-s + 36-s + 0.493·37-s + 0.162·38-s + 3.36·39-s + 1.56·41-s − 1.52·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4598\)    =    \(2 \cdot 11^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(36.7152\)
Root analytic conductor: \(6.05930\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4598,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.451465502\)
\(L(\frac12)\) \(\approx\) \(6.451465502\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
11 \( 1 \)
19 \( 1 - T \)
good3 \( 1 - p T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.336384672381450914953441469658, −7.68518050899319822002708251361, −6.89309234270293758384664065378, −6.26555728404467171168821694932, −5.25793582163031407488407859416, −4.15039718532399784162258574085, −3.81474771643367942690779784838, −2.99633193353632251635025171957, −2.20065862699872789756554872639, −1.34198304852947418622224891762, 1.34198304852947418622224891762, 2.20065862699872789756554872639, 2.99633193353632251635025171957, 3.81474771643367942690779784838, 4.15039718532399784162258574085, 5.25793582163031407488407859416, 6.26555728404467171168821694932, 6.89309234270293758384664065378, 7.68518050899319822002708251361, 8.336384672381450914953441469658

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