Properties

Label 2-4598-1.1-c1-0-10
Degree $2$
Conductor $4598$
Sign $1$
Analytic cond. $36.7152$
Root an. cond. $6.05930$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 3·7-s − 8-s − 3·9-s − 13-s + 3·14-s + 16-s − 5·17-s + 3·18-s − 19-s + 5·23-s − 5·25-s + 26-s − 3·28-s − 6·29-s + 2·31-s − 32-s + 5·34-s − 3·36-s − 3·37-s + 38-s + 2·41-s + 4·43-s − 5·46-s − 3·47-s + 2·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.13·7-s − 0.353·8-s − 9-s − 0.277·13-s + 0.801·14-s + 1/4·16-s − 1.21·17-s + 0.707·18-s − 0.229·19-s + 1.04·23-s − 25-s + 0.196·26-s − 0.566·28-s − 1.11·29-s + 0.359·31-s − 0.176·32-s + 0.857·34-s − 1/2·36-s − 0.493·37-s + 0.162·38-s + 0.312·41-s + 0.609·43-s − 0.737·46-s − 0.437·47-s + 2/7·49-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: yes
Arithmetic: yes
Character: $\chi_{4598} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4598,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4907690306\)
\(L(\frac12)\) \(\approx\) \(0.4907690306\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 15 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.581667113096120600802336877701, −7.54918850751004911454935889811, −6.97155050259421063428873086985, −6.18461518574718440667927582930, −5.68875401236409707623437779186, −4.58579598877322811410162405584, −3.52550885482380618682257711478, −2.80583407766484257278858832428, −1.97296683610578027538137202597, −0.40520920389872484228368834466, 0.40520920389872484228368834466, 1.97296683610578027538137202597, 2.80583407766484257278858832428, 3.52550885482380618682257711478, 4.58579598877322811410162405584, 5.68875401236409707623437779186, 6.18461518574718440667927582930, 6.97155050259421063428873086985, 7.54918850751004911454935889811, 8.581667113096120600802336877701

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