Properties

Label 2-4598-1.1-c1-0-138
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 + 3.30·3-s + 4-s + 1.88·5-s + 3.30·6-s + 2.41·7-s + 8-s + 7.91·9-s + 1.88·10-s + 3.30·12-s − 3.16·13-s + 2.41·14-s + 6.21·15-s + 16-s − 0.0684·17-s + 7.91·18-s − 19-s + 1.88·20-s + 7.96·21-s − 7.39·23-s + 3.30·24-s − 1.45·25-s − 3.16·26-s + 16.2·27-s + 2.41·28-s − 7.11·29-s + 6.21·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.90·3-s + 0.5·4-s + 0.841·5-s + 1.34·6-s + 0.911·7-s + 0.353·8-s + 2.63·9-s + 0.595·10-s + 0.953·12-s − 0.878·13-s + 0.644·14-s + 1.60·15-s + 0.250·16-s − 0.0165·17-s + 1.86·18-s − 0.229·19-s + 0.420·20-s + 1.73·21-s − 1.54·23-s + 0.674·24-s − 0.291·25-s − 0.620·26-s + 3.12·27-s + 0.455·28-s − 1.32·29-s + 1.13·30-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

\(L(1)\) \(\approx\) \(7.871214298\)
\(L(\frac12)\) \(\approx\) \(7.871214298\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.88T + 5T^{2} \)
7 \( 1 - 2.41T + 7T^{2} \)
13 \( 1 + 3.16T + 13T^{2} \)
17 \( 1 + 0.0684T + 17T^{2} \)
23 \( 1 + 7.39T + 23T^{2} \)
29 \( 1 + 7.11T + 29T^{2} \)
31 \( 1 + 5.53T + 31T^{2} \)
37 \( 1 - 8.71T + 37T^{2} \)
41 \( 1 - 2.36T + 41T^{2} \)
43 \( 1 + 1.76T + 43T^{2} \)
47 \( 1 - 9.11T + 47T^{2} \)
53 \( 1 - 8.63T + 53T^{2} \)
59 \( 1 - 5.32T + 59T^{2} \)
61 \( 1 + 14.3T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 + 13.1T + 71T^{2} \)
73 \( 1 - 5.34T + 73T^{2} \)
79 \( 1 + 3.95T + 79T^{2} \)
83 \( 1 + 5.25T + 83T^{2} \)
89 \( 1 - 6.41T + 89T^{2} \)
97 \( 1 + 2.67T + 97T^{2} \)
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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

−8.164246710933097129709280390068, −7.66603828099486349846310102542, −7.14505903211928730934429142039, −6.04135898429135781697202585648, −5.27782528241680920210465513770, −4.26566389933729212294214050304, −3.88702462174909922337956909987, −2.71770284150304984995287320186, −2.14280048152680081369307610879, −1.61710263310366811897825590658, 1.61710263310366811897825590658, 2.14280048152680081369307610879, 2.71770284150304984995287320186, 3.88702462174909922337956909987, 4.26566389933729212294214050304, 5.27782528241680920210465513770, 6.04135898429135781697202585648, 7.14505903211928730934429142039, 7.66603828099486349846310102542, 8.164246710933097129709280390068

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