Properties

Label 2-4598-1.1-c1-0-123
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 0.912·3-s + 4-s − 1.77·5-s − 0.912·6-s + 5.04·7-s − 8-s − 2.16·9-s + 1.77·10-s + 0.912·12-s − 5.42·13-s − 5.04·14-s − 1.62·15-s + 16-s + 1.82·17-s + 2.16·18-s + 19-s − 1.77·20-s + 4.60·21-s + 5.61·23-s − 0.912·24-s − 1.84·25-s + 5.42·26-s − 4.71·27-s + 5.04·28-s − 2.23·29-s + 1.62·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.527·3-s + 0.5·4-s − 0.794·5-s − 0.372·6-s + 1.90·7-s − 0.353·8-s − 0.722·9-s + 0.562·10-s + 0.263·12-s − 1.50·13-s − 1.34·14-s − 0.418·15-s + 0.250·16-s + 0.442·17-s + 0.510·18-s + 0.229·19-s − 0.397·20-s + 1.00·21-s + 1.17·23-s − 0.186·24-s − 0.368·25-s + 1.06·26-s − 0.907·27-s + 0.953·28-s − 0.415·29-s + 0.296·30-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 \( 1 - 0.912T + 3T^{2} \)
5 \( 1 + 1.77T + 5T^{2} \)
7 \( 1 - 5.04T + 7T^{2} \)
13 \( 1 + 5.42T + 13T^{2} \)
17 \( 1 - 1.82T + 17T^{2} \)
23 \( 1 - 5.61T + 23T^{2} \)
29 \( 1 + 2.23T + 29T^{2} \)
31 \( 1 - 0.386T + 31T^{2} \)
37 \( 1 + 10.0T + 37T^{2} \)
41 \( 1 + 9.28T + 41T^{2} \)
43 \( 1 + 5.45T + 43T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 - 6.85T + 53T^{2} \)
59 \( 1 - 3.70T + 59T^{2} \)
61 \( 1 + 2.89T + 61T^{2} \)
67 \( 1 + 11.5T + 67T^{2} \)
71 \( 1 + 0.872T + 71T^{2} \)
73 \( 1 + 9.78T + 73T^{2} \)
79 \( 1 + 9.31T + 79T^{2} \)
83 \( 1 - 4.28T + 83T^{2} \)
89 \( 1 + 5.60T + 89T^{2} \)
97 \( 1 + 2.90T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.957535315069225140390674567713, −7.48439083816546941922440321486, −7.02890727388534151889640548266, −5.49733819614987656645320138029, −5.11692080751253659325388970084, −4.17045708742979386695062889147, −3.15571016576172783087435960073, −2.29628110381337562636578656902, −1.42617288043383203285193359865, 0, 1.42617288043383203285193359865, 2.29628110381337562636578656902, 3.15571016576172783087435960073, 4.17045708742979386695062889147, 5.11692080751253659325388970084, 5.49733819614987656645320138029, 7.02890727388534151889640548266, 7.48439083816546941922440321486, 7.957535315069225140390674567713

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