Properties

Label 2-4998-1.1-c1-0-76
Degree $2$
Conductor $4998$
Sign $-1$
Analytic cond. $39.9092$
Root an. cond. $6.31737$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s − 2·11-s + 12-s − 13-s − 15-s + 16-s − 17-s − 18-s − 4·19-s − 20-s + 2·22-s + 8·23-s − 24-s − 4·25-s + 26-s + 27-s + 3·29-s + 30-s + 3·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.426·22-s + 1.66·23-s − 0.204·24-s − 4/5·25-s + 0.196·26-s + 0.192·27-s + 0.557·29-s + 0.182·30-s + 0.538·31-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: yes
Arithmetic: yes
Character: $\chi_{4998} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4998,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 - T \)
7 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−8.008511545080969112460235710317, −7.33044077059977358094987665447, −6.72810717696219618470036495320, −5.83981445251251161280485124985, −4.82265505809431164197107683788, −4.10849637293925602709498579647, −3.01982318488002607913305604355, −2.46453521363636310026877718975, −1.31275926206365428979706797643, 0, 1.31275926206365428979706797643, 2.46453521363636310026877718975, 3.01982318488002607913305604355, 4.10849637293925602709498579647, 4.82265505809431164197107683788, 5.83981445251251161280485124985, 6.72810717696219618470036495320, 7.33044077059977358094987665447, 8.008511545080969112460235710317

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