Properties

Label 2-4598-1.1-c1-0-127
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 + 2.48·3-s + 4-s + 1.48·5-s + 2.48·6-s + 4.90·7-s + 8-s + 3.18·9-s + 1.48·10-s + 2.48·12-s − 3.09·13-s + 4.90·14-s + 3.70·15-s + 16-s − 5.90·17-s + 3.18·18-s + 19-s + 1.48·20-s + 12.1·21-s + 2.09·23-s + 2.48·24-s − 2.78·25-s − 3.09·26-s + 0.468·27-s + 4.90·28-s − 0.700·29-s + 3.70·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.43·3-s + 0.5·4-s + 0.665·5-s + 1.01·6-s + 1.85·7-s + 0.353·8-s + 1.06·9-s + 0.470·10-s + 0.718·12-s − 0.857·13-s + 1.31·14-s + 0.955·15-s + 0.250·16-s − 1.43·17-s + 0.751·18-s + 0.229·19-s + 0.332·20-s + 2.66·21-s + 0.436·23-s + 0.507·24-s − 0.557·25-s − 0.606·26-s + 0.0900·27-s + 0.926·28-s − 0.130·29-s + 0.675·30-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

\(L(1)\) \(\approx\) \(6.941703207\)
\(L(\frac12)\) \(\approx\) \(6.941703207\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.48T + 3T^{2} \)
5 \( 1 - 1.48T + 5T^{2} \)
7 \( 1 - 4.90T + 7T^{2} \)
13 \( 1 + 3.09T + 13T^{2} \)
17 \( 1 + 5.90T + 17T^{2} \)
23 \( 1 - 2.09T + 23T^{2} \)
29 \( 1 + 0.700T + 29T^{2} \)
31 \( 1 - 4.97T + 31T^{2} \)
37 \( 1 - 9.31T + 37T^{2} \)
41 \( 1 + 6.48T + 41T^{2} \)
43 \( 1 - 0.792T + 43T^{2} \)
47 \( 1 + 2.18T + 47T^{2} \)
53 \( 1 + 5.29T + 53T^{2} \)
59 \( 1 + 0.603T + 59T^{2} \)
61 \( 1 - 5.11T + 61T^{2} \)
67 \( 1 + 11.8T + 67T^{2} \)
71 \( 1 - 9.45T + 71T^{2} \)
73 \( 1 - 1.90T + 73T^{2} \)
79 \( 1 - 14.4T + 79T^{2} \)
83 \( 1 + 7.48T + 83T^{2} \)
89 \( 1 - 3.50T + 89T^{2} \)
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

−8.178396162297631613157856639794, −7.76751668048463501075528547748, −7.00852046259641683365981854813, −6.07896903445151958285432444669, −5.04373487579619737031195554456, −4.64066644005231217420585661219, −3.83660789915673565219901422352, −2.63201622760380838020872472529, −2.23598363123536398039563528308, −1.46626578204086222463185624527, 1.46626578204086222463185624527, 2.23598363123536398039563528308, 2.63201622760380838020872472529, 3.83660789915673565219901422352, 4.64066644005231217420585661219, 5.04373487579619737031195554456, 6.07896903445151958285432444669, 7.00852046259641683365981854813, 7.76751668048463501075528547748, 8.178396162297631613157856639794

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