Properties

Label 2-4598-1.1-c1-0-134
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 + 1.43·3-s + 4-s − 0.301·5-s − 1.43·6-s + 7-s − 8-s − 0.952·9-s + 0.301·10-s + 1.43·12-s + 1.39·13-s − 14-s − 0.430·15-s + 16-s + 3.90·17-s + 0.952·18-s − 19-s − 0.301·20-s + 1.43·21-s − 3.57·23-s − 1.43·24-s − 4.90·25-s − 1.39·26-s − 5.65·27-s + 28-s + 0.635·29-s + 0.430·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.826·3-s + 0.5·4-s − 0.134·5-s − 0.584·6-s + 0.377·7-s − 0.353·8-s − 0.317·9-s + 0.0952·10-s + 0.413·12-s + 0.387·13-s − 0.267·14-s − 0.111·15-s + 0.250·16-s + 0.948·17-s + 0.224·18-s − 0.229·19-s − 0.0673·20-s + 0.312·21-s − 0.745·23-s − 0.292·24-s − 0.981·25-s − 0.274·26-s − 1.08·27-s + 0.188·28-s + 0.118·29-s + 0.0786·30-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 \( 1 - 1.43T + 3T^{2} \)
5 \( 1 + 0.301T + 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
13 \( 1 - 1.39T + 13T^{2} \)
17 \( 1 - 3.90T + 17T^{2} \)
23 \( 1 + 3.57T + 23T^{2} \)
29 \( 1 - 0.635T + 29T^{2} \)
31 \( 1 + 8.90T + 31T^{2} \)
37 \( 1 + 0.187T + 37T^{2} \)
41 \( 1 + 8.02T + 41T^{2} \)
43 \( 1 - 0.641T + 43T^{2} \)
47 \( 1 + 7.55T + 47T^{2} \)
53 \( 1 + 0.919T + 53T^{2} \)
59 \( 1 - 0.502T + 59T^{2} \)
61 \( 1 - 1.41T + 61T^{2} \)
67 \( 1 - 8.14T + 67T^{2} \)
71 \( 1 - 7.20T + 71T^{2} \)
73 \( 1 + 4.86T + 73T^{2} \)
79 \( 1 + 5.73T + 79T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 - 3.76T + 89T^{2} \)
97 \( 1 + 1.58T + 97T^{2} \)
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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

−8.033876870962830855705512647414, −7.63284662089222882704999703772, −6.66244832509795668427257559611, −5.83626149971570827405659741455, −5.11273713789340872070425600766, −3.81205859925938481124190076969, −3.34875909715844792522425004192, −2.24620497574495737774455986600, −1.53360365029604361466778599690, 0, 1.53360365029604361466778599690, 2.24620497574495737774455986600, 3.34875909715844792522425004192, 3.81205859925938481124190076969, 5.11273713789340872070425600766, 5.83626149971570827405659741455, 6.66244832509795668427257559611, 7.63284662089222882704999703772, 8.033876870962830855705512647414

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