Properties

Label 2-4998-1.1-c1-0-23
Degree $2$
Conductor $4998$
Sign $1$
Analytic cond. $39.9092$
Root an. cond. $6.31737$
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 − 3-s + 4-s − 1.46·5-s − 6-s + 8-s + 9-s − 1.46·10-s + 2.22·11-s − 12-s + 2.29·13-s + 1.46·15-s + 16-s − 17-s + 18-s + 4.07·19-s − 1.46·20-s + 2.22·22-s + 2.65·23-s − 24-s − 2.85·25-s + 2.29·26-s − 27-s + 8.90·29-s + 1.46·30-s − 2.07·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.655·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.463·10-s + 0.669·11-s − 0.288·12-s + 0.636·13-s + 0.378·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 0.934·19-s − 0.327·20-s + 0.473·22-s + 0.554·23-s − 0.204·24-s − 0.570·25-s + 0.449·26-s − 0.192·27-s + 1.65·29-s + 0.267·30-s − 0.372·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4998\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(39.9092\)
Root analytic conductor: \(6.31737\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4998,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.434071855\)
\(L(\frac12)\) \(\approx\) \(2.434071855\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
7 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + 1.46T + 5T^{2} \)
11 \( 1 - 2.22T + 11T^{2} \)
13 \( 1 - 2.29T + 13T^{2} \)
19 \( 1 - 4.07T + 19T^{2} \)
23 \( 1 - 2.65T + 23T^{2} \)
29 \( 1 - 8.90T + 29T^{2} \)
31 \( 1 + 2.07T + 31T^{2} \)
37 \( 1 - 0.364T + 37T^{2} \)
41 \( 1 + 7.83T + 41T^{2} \)
43 \( 1 + 4.91T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 - 4.09T + 53T^{2} \)
59 \( 1 + 6.16T + 59T^{2} \)
61 \( 1 + 2.38T + 61T^{2} \)
67 \( 1 - 1.28T + 67T^{2} \)
71 \( 1 - 9.17T + 71T^{2} \)
73 \( 1 + 0.191T + 73T^{2} \)
79 \( 1 - 8.99T + 79T^{2} \)
83 \( 1 - 16.2T + 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 - 12.5T + 97T^{2} \)
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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

−8.100078117095345913131421842412, −7.38346908588372771220242709815, −6.56328314407100862347250525953, −6.21356200043729627680077276537, −5.11217101042839349231217352648, −4.70350580141313980277962409727, −3.68267857391977724820763813154, −3.25488686184125629890773258909, −1.87127334835616362487249617970, −0.814767265504505984675732409762, 0.814767265504505984675732409762, 1.87127334835616362487249617970, 3.25488686184125629890773258909, 3.68267857391977724820763813154, 4.70350580141313980277962409727, 5.11217101042839349231217352648, 6.21356200043729627680077276537, 6.56328314407100862347250525953, 7.38346908588372771220242709815, 8.100078117095345913131421842412

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