Properties

Label 2-4598-1.1-c1-0-8
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 − 2.14·3-s + 4-s − 1.60·5-s − 2.14·6-s − 2.89·7-s + 8-s + 1.60·9-s − 1.60·10-s − 2.14·12-s − 4.89·13-s − 2.89·14-s + 3.43·15-s + 16-s + 5.74·17-s + 1.60·18-s + 19-s − 1.60·20-s + 6.20·21-s − 7.74·23-s − 2.14·24-s − 2.43·25-s − 4.89·26-s + 3.00·27-s − 2.89·28-s − 5.34·29-s + 3.43·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.23·3-s + 0.5·4-s − 0.716·5-s − 0.875·6-s − 1.09·7-s + 0.353·8-s + 0.533·9-s − 0.506·10-s − 0.619·12-s − 1.35·13-s − 0.772·14-s + 0.886·15-s + 0.250·16-s + 1.39·17-s + 0.377·18-s + 0.229·19-s − 0.358·20-s + 1.35·21-s − 1.61·23-s − 0.437·24-s − 0.487·25-s − 0.959·26-s + 0.577·27-s − 0.546·28-s − 0.993·29-s + 0.627·30-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

\(L(1)\) \(\approx\) \(0.6006735829\)
\(L(\frac12)\) \(\approx\) \(0.6006735829\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.14T + 3T^{2} \)
5 \( 1 + 1.60T + 5T^{2} \)
7 \( 1 + 2.89T + 7T^{2} \)
13 \( 1 + 4.89T + 13T^{2} \)
17 \( 1 - 5.74T + 17T^{2} \)
23 \( 1 + 7.74T + 23T^{2} \)
29 \( 1 + 5.34T + 29T^{2} \)
31 \( 1 + 7.43T + 31T^{2} \)
37 \( 1 + 7.49T + 37T^{2} \)
41 \( 1 - 2.05T + 41T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 + 7.08T + 47T^{2} \)
53 \( 1 - 8.32T + 53T^{2} \)
59 \( 1 + 2.54T + 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 - 14.9T + 71T^{2} \)
73 \( 1 - 3.74T + 73T^{2} \)
79 \( 1 - 8.69T + 79T^{2} \)
83 \( 1 - 6.10T + 83T^{2} \)
89 \( 1 - 3.20T + 89T^{2} \)
97 \( 1 + 6.98T + 97T^{2} \)
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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

−7.87672109323145568215943172123, −7.47089459622548593629933736718, −6.74213818813617001930405869079, −5.83525763078345636404303867827, −5.59083887467222899919622946134, −4.69687699668499012400286236153, −3.79734070238686811712256055200, −3.22074542336254436137027349507, −2.00291871005991870335858583302, −0.38907638635255401009910198760, 0.38907638635255401009910198760, 2.00291871005991870335858583302, 3.22074542336254436137027349507, 3.79734070238686811712256055200, 4.69687699668499012400286236153, 5.59083887467222899919622946134, 5.83525763078345636404303867827, 6.74213818813617001930405869079, 7.47089459622548593629933736718, 7.87672109323145568215943172123

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