Properties

Label 2-4598-1.1-c1-0-122
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 + 3.34·3-s + 4-s + 2.34·5-s − 3.34·6-s + 4.77·7-s − 8-s + 8.19·9-s − 2.34·10-s + 3.34·12-s − 1.57·13-s − 4.77·14-s + 7.85·15-s + 16-s − 3.77·17-s − 8.19·18-s − 19-s + 2.34·20-s + 15.9·21-s − 2.57·23-s − 3.34·24-s + 0.505·25-s + 1.57·26-s + 17.3·27-s + 4.77·28-s + 4.85·29-s − 7.85·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.93·3-s + 0.5·4-s + 1.04·5-s − 1.36·6-s + 1.80·7-s − 0.353·8-s + 2.73·9-s − 0.741·10-s + 0.965·12-s − 0.437·13-s − 1.27·14-s + 2.02·15-s + 0.250·16-s − 0.915·17-s − 1.93·18-s − 0.229·19-s + 0.524·20-s + 3.48·21-s − 0.537·23-s − 0.683·24-s + 0.101·25-s + 0.309·26-s + 3.34·27-s + 0.902·28-s + 0.900·29-s − 1.43·30-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

\(L(1)\) \(\approx\) \(4.465952007\)
\(L(\frac12)\) \(\approx\) \(4.465952007\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.34T + 3T^{2} \)
5 \( 1 - 2.34T + 5T^{2} \)
7 \( 1 - 4.77T + 7T^{2} \)
13 \( 1 + 1.57T + 13T^{2} \)
17 \( 1 + 3.77T + 17T^{2} \)
23 \( 1 + 2.57T + 23T^{2} \)
29 \( 1 - 4.85T + 29T^{2} \)
31 \( 1 - 6.69T + 31T^{2} \)
37 \( 1 + 10.8T + 37T^{2} \)
41 \( 1 + 7.00T + 41T^{2} \)
43 \( 1 + 11.8T + 43T^{2} \)
47 \( 1 + 7.19T + 47T^{2} \)
53 \( 1 + 1.14T + 53T^{2} \)
59 \( 1 - 4.92T + 59T^{2} \)
61 \( 1 - 4.04T + 61T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 + 2.31T + 71T^{2} \)
73 \( 1 - 7.77T + 73T^{2} \)
79 \( 1 + 0.992T + 79T^{2} \)
83 \( 1 - 8.34T + 83T^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−8.410293792625343962730689060979, −8.052506392252050819324583909344, −7.06735772465048582639621691858, −6.56633868167507943826104119567, −5.13916584474663823300265323969, −4.60260322193221114252244589636, −3.53581338379209747134411225842, −2.44280148020333503669290156801, −1.97280050037077129562492170259, −1.42486077276403594445916255565, 1.42486077276403594445916255565, 1.97280050037077129562492170259, 2.44280148020333503669290156801, 3.53581338379209747134411225842, 4.60260322193221114252244589636, 5.13916584474663823300265323969, 6.56633868167507943826104119567, 7.06735772465048582639621691858, 8.052506392252050819324583909344, 8.410293792625343962730689060979

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