Properties

Label 2-4598-1.1-c1-0-105
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 − 2·3-s + 4-s + 1.61·5-s − 2·6-s − 4.85·7-s + 8-s + 9-s + 1.61·10-s − 2·12-s + 2.47·13-s − 4.85·14-s − 3.23·15-s + 16-s + 3.38·17-s + 18-s + 19-s + 1.61·20-s + 9.70·21-s − 7.85·23-s − 2·24-s − 2.38·25-s + 2.47·26-s + 4·27-s − 4.85·28-s + 10.4·29-s − 3.23·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 0.5·4-s + 0.723·5-s − 0.816·6-s − 1.83·7-s + 0.353·8-s + 0.333·9-s + 0.511·10-s − 0.577·12-s + 0.685·13-s − 1.29·14-s − 0.835·15-s + 0.250·16-s + 0.820·17-s + 0.235·18-s + 0.229·19-s + 0.361·20-s + 2.11·21-s − 1.63·23-s − 0.408·24-s − 0.476·25-s + 0.484·26-s + 0.769·27-s − 0.917·28-s + 1.94·29-s − 0.590·30-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 \( 1 + 2T + 3T^{2} \)
5 \( 1 - 1.61T + 5T^{2} \)
7 \( 1 + 4.85T + 7T^{2} \)
13 \( 1 - 2.47T + 13T^{2} \)
17 \( 1 - 3.38T + 17T^{2} \)
23 \( 1 + 7.85T + 23T^{2} \)
29 \( 1 - 10.4T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 4.76T + 41T^{2} \)
43 \( 1 + 9.09T + 43T^{2} \)
47 \( 1 - 1.38T + 47T^{2} \)
53 \( 1 + 4.76T + 53T^{2} \)
59 \( 1 + 3.52T + 59T^{2} \)
61 \( 1 + 9.56T + 61T^{2} \)
67 \( 1 - 2.76T + 67T^{2} \)
71 \( 1 - 4.94T + 71T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 - 3.70T + 79T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 + 9.23T + 97T^{2} \)
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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

−7.78170163991278295346999971229, −6.61837419657795573612745364479, −6.34264032383615264793978943191, −5.91331107784492094247068093318, −5.24876326650737583986214761280, −4.25174262966563049299176686504, −3.35575911344486109989536598598, −2.68148181105250630854910565410, −1.30808828819426220615574303115, 0, 1.30808828819426220615574303115, 2.68148181105250630854910565410, 3.35575911344486109989536598598, 4.25174262966563049299176686504, 5.24876326650737583986214761280, 5.91331107784492094247068093318, 6.34264032383615264793978943191, 6.61837419657795573612745364479, 7.78170163991278295346999971229

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