Properties

Label 2-4598-1.1-c1-0-73
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 − 0.698·3-s + 4-s − 2.43·5-s + 0.698·6-s + 7-s − 8-s − 2.51·9-s + 2.43·10-s − 0.698·12-s − 2.86·13-s − 14-s + 1.69·15-s + 16-s − 1.90·17-s + 2.51·18-s − 19-s − 2.43·20-s − 0.698·21-s + 6.50·23-s + 0.698·24-s + 0.909·25-s + 2.86·26-s + 3.85·27-s + 28-s + 7.02·29-s − 1.69·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.403·3-s + 0.5·4-s − 1.08·5-s + 0.285·6-s + 0.377·7-s − 0.353·8-s − 0.837·9-s + 0.768·10-s − 0.201·12-s − 0.793·13-s − 0.267·14-s + 0.438·15-s + 0.250·16-s − 0.463·17-s + 0.591·18-s − 0.229·19-s − 0.543·20-s − 0.152·21-s + 1.35·23-s + 0.142·24-s + 0.181·25-s + 0.561·26-s + 0.741·27-s + 0.188·28-s + 1.30·29-s − 0.310·30-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 \( 1 + 0.698T + 3T^{2} \)
5 \( 1 + 2.43T + 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
13 \( 1 + 2.86T + 13T^{2} \)
17 \( 1 + 1.90T + 17T^{2} \)
23 \( 1 - 6.50T + 23T^{2} \)
29 \( 1 - 7.02T + 29T^{2} \)
31 \( 1 - 5.43T + 31T^{2} \)
37 \( 1 - 0.383T + 37T^{2} \)
41 \( 1 - 7.29T + 41T^{2} \)
43 \( 1 + 5.17T + 43T^{2} \)
47 \( 1 - 4.08T + 47T^{2} \)
53 \( 1 + 0.348T + 53T^{2} \)
59 \( 1 + 1.62T + 59T^{2} \)
61 \( 1 - 15.1T + 61T^{2} \)
67 \( 1 + 7.75T + 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 - 8.33T + 73T^{2} \)
79 \( 1 + 2.46T + 79T^{2} \)
83 \( 1 + 3.11T + 83T^{2} \)
89 \( 1 + 3.03T + 89T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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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

−8.234999307339322360131246502665, −7.25070775636492239351834471115, −6.75718347340221208926542011171, −5.84991464813146309409342755014, −4.92510652540056605999537513745, −4.33528591138150974948436921344, −3.13009240365442165388728266235, −2.47328869430187173104693663807, −1.00975746967924539269192426675, 0, 1.00975746967924539269192426675, 2.47328869430187173104693663807, 3.13009240365442165388728266235, 4.33528591138150974948436921344, 4.92510652540056605999537513745, 5.84991464813146309409342755014, 6.75718347340221208926542011171, 7.25070775636492239351834471115, 8.234999307339322360131246502665

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