Properties

Label 2-4598-1.1-c1-0-137
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 + 2.30·3-s + 4-s + 2.62·5-s + 2.30·6-s + 2.74·7-s + 8-s + 2.29·9-s + 2.62·10-s + 2.30·12-s + 5.41·13-s + 2.74·14-s + 6.04·15-s + 16-s − 4.27·17-s + 2.29·18-s − 19-s + 2.62·20-s + 6.30·21-s − 6.21·23-s + 2.30·24-s + 1.89·25-s + 5.41·26-s − 1.62·27-s + 2.74·28-s + 5.99·29-s + 6.04·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.32·3-s + 0.5·4-s + 1.17·5-s + 0.939·6-s + 1.03·7-s + 0.353·8-s + 0.764·9-s + 0.830·10-s + 0.664·12-s + 1.50·13-s + 0.732·14-s + 1.56·15-s + 0.250·16-s − 1.03·17-s + 0.540·18-s − 0.229·19-s + 0.587·20-s + 1.37·21-s − 1.29·23-s + 0.469·24-s + 0.379·25-s + 1.06·26-s − 0.312·27-s + 0.518·28-s + 1.11·29-s + 1.10·30-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4598,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.066407457\)
\(L(\frac12)\) \(\approx\) \(7.066407457\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.30T + 3T^{2} \)
5 \( 1 - 2.62T + 5T^{2} \)
7 \( 1 - 2.74T + 7T^{2} \)
13 \( 1 - 5.41T + 13T^{2} \)
17 \( 1 + 4.27T + 17T^{2} \)
23 \( 1 + 6.21T + 23T^{2} \)
29 \( 1 - 5.99T + 29T^{2} \)
31 \( 1 + 7.27T + 31T^{2} \)
37 \( 1 + 3.97T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + 0.306T + 43T^{2} \)
47 \( 1 + 4.56T + 47T^{2} \)
53 \( 1 + 3.72T + 53T^{2} \)
59 \( 1 - 5.42T + 59T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 + 0.0148T + 67T^{2} \)
71 \( 1 - 12.1T + 71T^{2} \)
73 \( 1 + 13.8T + 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 - 1.29T + 83T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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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

−8.279194157599047242708739371059, −7.82375347785358986559150617466, −6.73063497000463132460986057365, −6.06971000076131955875742613960, −5.41586882925337512042559241396, −4.39511374482008010303810474877, −3.81210463493006505711793540960, −2.82155736656073628172541221733, −1.99515226501266252970578461724, −1.56822291443605548787046138208, 1.56822291443605548787046138208, 1.99515226501266252970578461724, 2.82155736656073628172541221733, 3.81210463493006505711793540960, 4.39511374482008010303810474877, 5.41586882925337512042559241396, 6.06971000076131955875742613960, 6.73063497000463132460986057365, 7.82375347785358986559150617466, 8.279194157599047242708739371059

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