Properties

Label 2-4598-1.1-c1-0-129
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3.30·3-s + 4-s + 1.30·5-s − 3.30·6-s + 2.30·7-s + 8-s + 7.90·9-s + 1.30·10-s − 3.30·12-s − 0.302·13-s + 2.30·14-s − 4.30·15-s + 16-s − 2.60·17-s + 7.90·18-s − 19-s + 1.30·20-s − 7.60·21-s − 8.60·23-s − 3.30·24-s − 3.30·25-s − 0.302·26-s − 16.2·27-s + 2.30·28-s + 4.69·29-s − 4.30·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.90·3-s + 0.5·4-s + 0.582·5-s − 1.34·6-s + 0.870·7-s + 0.353·8-s + 2.63·9-s + 0.411·10-s − 0.953·12-s − 0.0839·13-s + 0.615·14-s − 1.11·15-s + 0.250·16-s − 0.631·17-s + 1.86·18-s − 0.229·19-s + 0.291·20-s − 1.65·21-s − 1.79·23-s − 0.674·24-s − 0.660·25-s − 0.0593·26-s − 3.11·27-s + 0.435·28-s + 0.872·29-s − 0.785·30-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: \(1\)
Selberg data: \((2,\ 4598,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 \)
19 \( 1 + T \)
good3 \( 1 + 3.30T + 3T^{2} \)
5 \( 1 - 1.30T + 5T^{2} \)
7 \( 1 - 2.30T + 7T^{2} \)
13 \( 1 + 0.302T + 13T^{2} \)
17 \( 1 + 2.60T + 17T^{2} \)
23 \( 1 + 8.60T + 23T^{2} \)
29 \( 1 - 4.69T + 29T^{2} \)
31 \( 1 - 0.302T + 31T^{2} \)
37 \( 1 + 9.21T + 37T^{2} \)
41 \( 1 - 6.90T + 41T^{2} \)
43 \( 1 + 11.9T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 3.39T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 3.21T + 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 + 4.60T + 73T^{2} \)
79 \( 1 - 5.81T + 79T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 + 2.60T + 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−7.69851203475497045286173525712, −6.85098900121693001421745282720, −6.25515810304723313731381613557, −5.77147233262827214693079606726, −4.98774941147231156275822313531, −4.58830060096391877825894587037, −3.73736176759141151028476772946, −2.09476324187039181284348221417, −1.48989443334990104529489706532, 0, 1.48989443334990104529489706532, 2.09476324187039181284348221417, 3.73736176759141151028476772946, 4.58830060096391877825894587037, 4.98774941147231156275822313531, 5.77147233262827214693079606726, 6.25515810304723313731381613557, 6.85098900121693001421745282720, 7.69851203475497045286173525712

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