Properties

Label 2-4598-1.1-c1-0-96
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 + 2.94·3-s + 4-s + 2.36·5-s − 2.94·6-s + 1.66·7-s − 8-s + 5.68·9-s − 2.36·10-s + 2.94·12-s + 0.633·13-s − 1.66·14-s + 6.97·15-s + 16-s − 0.581·17-s − 5.68·18-s + 19-s + 2.36·20-s + 4.89·21-s + 3.60·23-s − 2.94·24-s + 0.599·25-s − 0.633·26-s + 7.92·27-s + 1.66·28-s − 9.13·29-s − 6.97·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.70·3-s + 0.5·4-s + 1.05·5-s − 1.20·6-s + 0.627·7-s − 0.353·8-s + 1.89·9-s − 0.748·10-s + 0.850·12-s + 0.175·13-s − 0.443·14-s + 1.80·15-s + 0.250·16-s − 0.140·17-s − 1.34·18-s + 0.229·19-s + 0.529·20-s + 1.06·21-s + 0.752·23-s − 0.601·24-s + 0.119·25-s − 0.124·26-s + 1.52·27-s + 0.313·28-s − 1.69·29-s − 1.27·30-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

\(L(1)\) \(\approx\) \(3.760460322\)
\(L(\frac12)\) \(\approx\) \(3.760460322\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.94T + 3T^{2} \)
5 \( 1 - 2.36T + 5T^{2} \)
7 \( 1 - 1.66T + 7T^{2} \)
13 \( 1 - 0.633T + 13T^{2} \)
17 \( 1 + 0.581T + 17T^{2} \)
23 \( 1 - 3.60T + 23T^{2} \)
29 \( 1 + 9.13T + 29T^{2} \)
31 \( 1 - 1.02T + 31T^{2} \)
37 \( 1 - 2.15T + 37T^{2} \)
41 \( 1 - 4.40T + 41T^{2} \)
43 \( 1 - 9.58T + 43T^{2} \)
47 \( 1 - 8.73T + 47T^{2} \)
53 \( 1 + 12.5T + 53T^{2} \)
59 \( 1 - 6.47T + 59T^{2} \)
61 \( 1 - 1.32T + 61T^{2} \)
67 \( 1 + 0.393T + 67T^{2} \)
71 \( 1 + 0.312T + 71T^{2} \)
73 \( 1 + 9.31T + 73T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 + 6.90T + 83T^{2} \)
89 \( 1 - 0.519T + 89T^{2} \)
97 \( 1 + 8.37T + 97T^{2} \)
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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

−8.489958139492986179259568136325, −7.61785120657212534721617094064, −7.36903497650728595096402178348, −6.26573116865837367742697340582, −5.49333981594572747888659538258, −4.41851643715058228194144217155, −3.49937122741865751543156669854, −2.58706125201397096783267813807, −2.01133049881037536941929459089, −1.22976112593117943035176863945, 1.22976112593117943035176863945, 2.01133049881037536941929459089, 2.58706125201397096783267813807, 3.49937122741865751543156669854, 4.41851643715058228194144217155, 5.49333981594572747888659538258, 6.26573116865837367742697340582, 7.36903497650728595096402178348, 7.61785120657212534721617094064, 8.489958139492986179259568136325

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