Properties

Label 2-4598-1.1-c1-0-142
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 + 2.12·3-s + 4-s − 0.444·5-s − 2.12·6-s + 0.813·7-s − 8-s + 1.53·9-s + 0.444·10-s + 2.12·12-s + 0.531·13-s − 0.813·14-s − 0.946·15-s + 16-s − 5.88·17-s − 1.53·18-s − 19-s − 0.444·20-s + 1.73·21-s + 6.69·23-s − 2.12·24-s − 4.80·25-s − 0.531·26-s − 3.12·27-s + 0.813·28-s − 4.16·29-s + 0.946·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.22·3-s + 0.5·4-s − 0.198·5-s − 0.869·6-s + 0.307·7-s − 0.353·8-s + 0.510·9-s + 0.140·10-s + 0.614·12-s + 0.147·13-s − 0.217·14-s − 0.244·15-s + 0.250·16-s − 1.42·17-s − 0.360·18-s − 0.229·19-s − 0.0994·20-s + 0.378·21-s + 1.39·23-s − 0.434·24-s − 0.960·25-s − 0.104·26-s − 0.601·27-s + 0.153·28-s − 0.773·29-s + 0.172·30-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 \( 1 - 2.12T + 3T^{2} \)
5 \( 1 + 0.444T + 5T^{2} \)
7 \( 1 - 0.813T + 7T^{2} \)
13 \( 1 - 0.531T + 13T^{2} \)
17 \( 1 + 5.88T + 17T^{2} \)
23 \( 1 - 6.69T + 23T^{2} \)
29 \( 1 + 4.16T + 29T^{2} \)
31 \( 1 + 5.80T + 31T^{2} \)
37 \( 1 - 2.37T + 37T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 - 6.33T + 47T^{2} \)
53 \( 1 - 9.31T + 53T^{2} \)
59 \( 1 - 3.37T + 59T^{2} \)
61 \( 1 + 4.76T + 61T^{2} \)
67 \( 1 + 1.23T + 67T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 + 3.46T + 79T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 - 7.40T + 89T^{2} \)
97 \( 1 + 14.2T + 97T^{2} \)
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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

−8.219239979769288439043308646945, −7.33710859378639568736608906390, −6.92098131975446002089067485096, −5.88641030321470845932358796596, −4.91337325768479948353145537192, −3.92168203015765784775500243998, −3.21721555650249476053745248713, −2.28292448104231810278364958309, −1.62911797152089149980957944918, 0, 1.62911797152089149980957944918, 2.28292448104231810278364958309, 3.21721555650249476053745248713, 3.92168203015765784775500243998, 4.91337325768479948353145537192, 5.88641030321470845932358796596, 6.92098131975446002089067485096, 7.33710859378639568736608906390, 8.219239979769288439043308646945

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