Properties

Label 2-4598-1.1-c1-0-67
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3.05·3-s + 4-s − 1.32·5-s + 3.05·6-s + 7-s − 8-s + 6.35·9-s + 1.32·10-s − 3.05·12-s − 0.653·13-s − 14-s + 4.05·15-s + 16-s + 2.23·17-s − 6.35·18-s − 19-s − 1.32·20-s − 3.05·21-s − 3.31·23-s + 3.05·24-s − 3.23·25-s + 0.653·26-s − 10.2·27-s + 28-s + 0.248·29-s − 4.05·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.76·3-s + 0.5·4-s − 0.593·5-s + 1.24·6-s + 0.377·7-s − 0.353·8-s + 2.11·9-s + 0.419·10-s − 0.883·12-s − 0.181·13-s − 0.267·14-s + 1.04·15-s + 0.250·16-s + 0.543·17-s − 1.49·18-s − 0.229·19-s − 0.296·20-s − 0.667·21-s − 0.691·23-s + 0.624·24-s − 0.647·25-s + 0.128·26-s − 1.97·27-s + 0.188·28-s + 0.0461·29-s − 0.741·30-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: \(1\)
Selberg data: \((2,\ 4598,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.05T + 3T^{2} \)
5 \( 1 + 1.32T + 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
13 \( 1 + 0.653T + 13T^{2} \)
17 \( 1 - 2.23T + 17T^{2} \)
23 \( 1 + 3.31T + 23T^{2} \)
29 \( 1 - 0.248T + 29T^{2} \)
31 \( 1 - 7.26T + 31T^{2} \)
37 \( 1 - 1.22T + 37T^{2} \)
41 \( 1 + 4.04T + 41T^{2} \)
43 \( 1 + 4.49T + 43T^{2} \)
47 \( 1 + 0.746T + 47T^{2} \)
53 \( 1 - 3.95T + 53T^{2} \)
59 \( 1 - 9.86T + 59T^{2} \)
61 \( 1 + 8.07T + 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 - 5.01T + 71T^{2} \)
73 \( 1 + 8.83T + 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 - 8.49T + 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−7.85451404044703453598046841439, −7.25759026960460183913747835890, −6.42376004530200796192497885348, −5.95226531914287664176739347349, −5.05259037807373650839831161833, −4.45147854529290862801050479531, −3.46114613799644296860665328344, −2.00535456403352980783509387756, −0.970249398181954931609648358259, 0, 0.970249398181954931609648358259, 2.00535456403352980783509387756, 3.46114613799644296860665328344, 4.45147854529290862801050479531, 5.05259037807373650839831161833, 5.95226531914287664176739347349, 6.42376004530200796192497885348, 7.25759026960460183913747835890, 7.85451404044703453598046841439

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