Properties

Label 2-4598-1.1-c1-0-47
Degree $2$
Conductor $4598$
Sign $1$
Analytic cond. $36.7152$
Root an. cond. $6.05930$
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  + 2-s − 3-s + 4-s − 0.732·5-s − 6-s − 0.267·7-s + 8-s − 2·9-s − 0.732·10-s − 12-s + 3.46·13-s − 0.267·14-s + 0.732·15-s + 16-s + 7.46·17-s − 2·18-s − 19-s − 0.732·20-s + 0.267·21-s − 23-s − 24-s − 4.46·25-s + 3.46·26-s + 5·27-s − 0.267·28-s − 1.73·29-s + 0.732·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.327·5-s − 0.408·6-s − 0.101·7-s + 0.353·8-s − 0.666·9-s − 0.231·10-s − 0.288·12-s + 0.960·13-s − 0.0716·14-s + 0.189·15-s + 0.250·16-s + 1.81·17-s − 0.471·18-s − 0.229·19-s − 0.163·20-s + 0.0584·21-s − 0.208·23-s − 0.204·24-s − 0.892·25-s + 0.679·26-s + 0.962·27-s − 0.0506·28-s − 0.321·29-s + 0.133·30-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4598,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.202433125\)
\(L(\frac12)\) \(\approx\) \(2.202433125\)
\(L(\frac{3}{2})\) 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 \( 1 + T + 3T^{2} \)
5 \( 1 + 0.732T + 5T^{2} \)
7 \( 1 + 0.267T + 7T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 - 7.46T + 17T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 + 1.73T + 29T^{2} \)
31 \( 1 + 1.46T + 31T^{2} \)
37 \( 1 + 6.46T + 37T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
43 \( 1 + 4.92T + 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 + 9.92T + 53T^{2} \)
59 \( 1 + 9.39T + 59T^{2} \)
61 \( 1 + 1.26T + 61T^{2} \)
67 \( 1 - 14T + 67T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 2.19T + 79T^{2} \)
83 \( 1 - 3.80T + 83T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 - 13.4T + 97T^{2} \)
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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

−8.048442498532947110226646357535, −7.66229978519197040150384382025, −6.60814751268699454778574233639, −5.93307109981927204350756998896, −5.53748924781498567311298036389, −4.68297727645734456090207918537, −3.64260003218086507290713055696, −3.26374825631749609900646047872, −1.99204649069875756022141574344, −0.76657129701112806050763467212, 0.76657129701112806050763467212, 1.99204649069875756022141574344, 3.26374825631749609900646047872, 3.64260003218086507290713055696, 4.68297727645734456090207918537, 5.53748924781498567311298036389, 5.93307109981927204350756998896, 6.60814751268699454778574233639, 7.66229978519197040150384382025, 8.048442498532947110226646357535

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