Properties

Label 2-4598-1.1-c1-0-44
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 + 0.180·3-s + 4-s + 4.08·5-s − 0.180·6-s − 3.52·7-s − 8-s − 2.96·9-s − 4.08·10-s + 0.180·12-s − 1.08·13-s + 3.52·14-s + 0.739·15-s + 16-s + 3.90·17-s + 2.96·18-s + 19-s + 4.08·20-s − 0.638·21-s − 4.34·23-s − 0.180·24-s + 11.6·25-s + 1.08·26-s − 1.07·27-s − 3.52·28-s + 6.06·29-s − 0.739·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.104·3-s + 0.5·4-s + 1.82·5-s − 0.0738·6-s − 1.33·7-s − 0.353·8-s − 0.989·9-s − 1.29·10-s + 0.0522·12-s − 0.300·13-s + 0.942·14-s + 0.190·15-s + 0.250·16-s + 0.946·17-s + 0.699·18-s + 0.229·19-s + 0.913·20-s − 0.139·21-s − 0.905·23-s − 0.0369·24-s + 2.33·25-s + 0.212·26-s − 0.207·27-s − 0.666·28-s + 1.12·29-s − 0.134·30-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

\(L(1)\) \(\approx\) \(1.559652653\)
\(L(\frac12)\) \(\approx\) \(1.559652653\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.180T + 3T^{2} \)
5 \( 1 - 4.08T + 5T^{2} \)
7 \( 1 + 3.52T + 7T^{2} \)
13 \( 1 + 1.08T + 13T^{2} \)
17 \( 1 - 3.90T + 17T^{2} \)
23 \( 1 + 4.34T + 23T^{2} \)
29 \( 1 - 6.06T + 29T^{2} \)
31 \( 1 - 7.26T + 31T^{2} \)
37 \( 1 - 0.754T + 37T^{2} \)
41 \( 1 + 6.67T + 41T^{2} \)
43 \( 1 + 4.60T + 43T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 - 4.42T + 53T^{2} \)
59 \( 1 + 3.54T + 59T^{2} \)
61 \( 1 + 9.05T + 61T^{2} \)
67 \( 1 - 1.35T + 67T^{2} \)
71 \( 1 + 8.96T + 71T^{2} \)
73 \( 1 + 8.26T + 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 - 17.6T + 83T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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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

−8.612016353462788929120721517771, −7.63795299245119141112779315255, −6.71269482379885776474489274249, −6.05977360400240271099325508969, −5.84032017611929325412434763767, −4.83231541950650333474401526495, −3.26828654519681593572344130523, −2.79680450110051065898305431276, −1.96566334169124693427896503944, −0.75278835288491543619475530899, 0.75278835288491543619475530899, 1.96566334169124693427896503944, 2.79680450110051065898305431276, 3.26828654519681593572344130523, 4.83231541950650333474401526495, 5.84032017611929325412434763767, 6.05977360400240271099325508969, 6.71269482379885776474489274249, 7.63795299245119141112779315255, 8.612016353462788929120721517771

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