Properties

Label 2-4598-1.1-c1-0-6
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: $\chi_{4598} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4598,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3362056027\)
\(L(\frac12)\) \(\approx\) \(0.3362056027\)
\(L(\frac{3}{2})\) 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

−8.321097595735992382980196174602, −7.68840621609576732569336126168, −6.82272702198640737011776094892, −6.37262799311460060404166761712, −5.38266803331276432754755090571, −4.76852537712280093870554148054, −3.74924779892313023992806743840, −2.66527301920239186644644347413, −1.87687307597278187780052376503, −0.34922137815240737037392207977, 0.34922137815240737037392207977, 1.87687307597278187780052376503, 2.66527301920239186644644347413, 3.74924779892313023992806743840, 4.76852537712280093870554148054, 5.38266803331276432754755090571, 6.37262799311460060404166761712, 6.82272702198640737011776094892, 7.68840621609576732569336126168, 8.321097595735992382980196174602

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