Properties

Label 2-8042-1.1-c1-0-97
Degree $2$
Conductor $8042$
Sign $1$
Analytic cond. $64.2156$
Root an. cond. $8.01346$
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 − 2.38·3-s + 4-s + 2.54·5-s + 2.38·6-s + 3.05·7-s − 8-s + 2.69·9-s − 2.54·10-s + 5.40·11-s − 2.38·12-s + 2.16·13-s − 3.05·14-s − 6.06·15-s + 16-s + 0.860·17-s − 2.69·18-s − 6.33·19-s + 2.54·20-s − 7.28·21-s − 5.40·22-s − 2.60·23-s + 2.38·24-s + 1.46·25-s − 2.16·26-s + 0.739·27-s + 3.05·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.37·3-s + 0.5·4-s + 1.13·5-s + 0.973·6-s + 1.15·7-s − 0.353·8-s + 0.896·9-s − 0.803·10-s + 1.63·11-s − 0.688·12-s + 0.599·13-s − 0.816·14-s − 1.56·15-s + 0.250·16-s + 0.208·17-s − 0.634·18-s − 1.45·19-s + 0.568·20-s − 1.59·21-s − 1.15·22-s − 0.542·23-s + 0.486·24-s + 0.292·25-s − 0.423·26-s + 0.142·27-s + 0.577·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8042\)    =    \(2 \cdot 4021\)
Sign: $1$
Analytic conductor: \(64.2156\)
Root analytic conductor: \(8.01346\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8042,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.403258798\)
\(L(\frac12)\) \(\approx\) \(1.403258798\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
4021 \( 1+O(T) \)
good3 \( 1 + 2.38T + 3T^{2} \)
5 \( 1 - 2.54T + 5T^{2} \)
7 \( 1 - 3.05T + 7T^{2} \)
11 \( 1 - 5.40T + 11T^{2} \)
13 \( 1 - 2.16T + 13T^{2} \)
17 \( 1 - 0.860T + 17T^{2} \)
19 \( 1 + 6.33T + 19T^{2} \)
23 \( 1 + 2.60T + 23T^{2} \)
29 \( 1 + 9.11T + 29T^{2} \)
31 \( 1 - 5.79T + 31T^{2} \)
37 \( 1 + 3.68T + 37T^{2} \)
41 \( 1 + 2.55T + 41T^{2} \)
43 \( 1 + 2.74T + 43T^{2} \)
47 \( 1 + 8.15T + 47T^{2} \)
53 \( 1 - 13.6T + 53T^{2} \)
59 \( 1 - 4.92T + 59T^{2} \)
61 \( 1 - 8.90T + 61T^{2} \)
67 \( 1 + 8.71T + 67T^{2} \)
71 \( 1 + 5.52T + 71T^{2} \)
73 \( 1 + 1.03T + 73T^{2} \)
79 \( 1 - 15.5T + 79T^{2} \)
83 \( 1 - 6.14T + 83T^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−7.87433212832950000250539549473, −6.85446417126600161510516220524, −6.40699663486765031473163620466, −5.91130045429054122161840123091, −5.28558923925155340290156807155, −4.46109284197093333483090408207, −3.65424137583157741645575134080, −2.00905119263080677945380198717, −1.68806621939473063487071328002, −0.73930823400713310815092294811, 0.73930823400713310815092294811, 1.68806621939473063487071328002, 2.00905119263080677945380198717, 3.65424137583157741645575134080, 4.46109284197093333483090408207, 5.28558923925155340290156807155, 5.91130045429054122161840123091, 6.40699663486765031473163620466, 6.85446417126600161510516220524, 7.87433212832950000250539549473

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