Properties

Label 2-4598-1.1-c1-0-91
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 + 1.67·3-s + 4-s − 0.566·5-s + 1.67·6-s + 3.40·7-s + 8-s − 0.199·9-s − 0.566·10-s + 1.67·12-s + 1.35·13-s + 3.40·14-s − 0.948·15-s + 16-s + 4.30·17-s − 0.199·18-s − 19-s − 0.566·20-s + 5.69·21-s − 2.71·23-s + 1.67·24-s − 4.67·25-s + 1.35·26-s − 5.35·27-s + 3.40·28-s + 5.64·29-s − 0.948·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.966·3-s + 0.5·4-s − 0.253·5-s + 0.683·6-s + 1.28·7-s + 0.353·8-s − 0.0664·9-s − 0.179·10-s + 0.483·12-s + 0.375·13-s + 0.908·14-s − 0.244·15-s + 0.250·16-s + 1.04·17-s − 0.0469·18-s − 0.229·19-s − 0.126·20-s + 1.24·21-s − 0.565·23-s + 0.341·24-s − 0.935·25-s + 0.265·26-s − 1.03·27-s + 0.642·28-s + 1.04·29-s − 0.173·30-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

\(L(1)\) \(\approx\) \(5.046718211\)
\(L(\frac12)\) \(\approx\) \(5.046718211\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.67T + 3T^{2} \)
5 \( 1 + 0.566T + 5T^{2} \)
7 \( 1 - 3.40T + 7T^{2} \)
13 \( 1 - 1.35T + 13T^{2} \)
17 \( 1 - 4.30T + 17T^{2} \)
23 \( 1 + 2.71T + 23T^{2} \)
29 \( 1 - 5.64T + 29T^{2} \)
31 \( 1 - 9.62T + 31T^{2} \)
37 \( 1 - 3.63T + 37T^{2} \)
41 \( 1 + 3.64T + 41T^{2} \)
43 \( 1 + 7.94T + 43T^{2} \)
47 \( 1 - 13.5T + 47T^{2} \)
53 \( 1 - 4.25T + 53T^{2} \)
59 \( 1 - 7.97T + 59T^{2} \)
61 \( 1 - 9.37T + 61T^{2} \)
67 \( 1 - 1.01T + 67T^{2} \)
71 \( 1 - 6.30T + 71T^{2} \)
73 \( 1 + 14.0T + 73T^{2} \)
79 \( 1 + 7.11T + 79T^{2} \)
83 \( 1 + 1.02T + 83T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 - 7.04T + 97T^{2} \)
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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

−8.279877031665342665604170372821, −7.80322297065595968216663254420, −6.96619875614823865872347626348, −5.95356968097107827814997020372, −5.33975774143284671681356588616, −4.40160810722695009920330954667, −3.85051205161488952541515971454, −2.91472850206919615310001540957, −2.19230459402056782753035371882, −1.16994541459959863591064273553, 1.16994541459959863591064273553, 2.19230459402056782753035371882, 2.91472850206919615310001540957, 3.85051205161488952541515971454, 4.40160810722695009920330954667, 5.33975774143284671681356588616, 5.95356968097107827814997020372, 6.96619875614823865872347626348, 7.80322297065595968216663254420, 8.279877031665342665604170372821

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