Properties

Label 2-8042-1.1-c1-0-49
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 − 2.59·3-s + 4-s − 0.859·5-s + 2.59·6-s − 3.29·7-s − 8-s + 3.72·9-s + 0.859·10-s − 0.214·11-s − 2.59·12-s + 0.577·13-s + 3.29·14-s + 2.22·15-s + 16-s + 6.01·17-s − 3.72·18-s + 0.272·19-s − 0.859·20-s + 8.54·21-s + 0.214·22-s + 2.84·23-s + 2.59·24-s − 4.26·25-s − 0.577·26-s − 1.88·27-s − 3.29·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.49·3-s + 0.5·4-s − 0.384·5-s + 1.05·6-s − 1.24·7-s − 0.353·8-s + 1.24·9-s + 0.271·10-s − 0.0646·11-s − 0.748·12-s + 0.160·13-s + 0.880·14-s + 0.575·15-s + 0.250·16-s + 1.45·17-s − 0.878·18-s + 0.0624·19-s − 0.192·20-s + 1.86·21-s + 0.0457·22-s + 0.592·23-s + 0.529·24-s − 0.852·25-s − 0.113·26-s − 0.363·27-s − 0.622·28-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

\(L(1)\) \(\approx\) \(0.5242889581\)
\(L(\frac12)\) \(\approx\) \(0.5242889581\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.59T + 3T^{2} \)
5 \( 1 + 0.859T + 5T^{2} \)
7 \( 1 + 3.29T + 7T^{2} \)
11 \( 1 + 0.214T + 11T^{2} \)
13 \( 1 - 0.577T + 13T^{2} \)
17 \( 1 - 6.01T + 17T^{2} \)
19 \( 1 - 0.272T + 19T^{2} \)
23 \( 1 - 2.84T + 23T^{2} \)
29 \( 1 - 3.81T + 29T^{2} \)
31 \( 1 - 8.62T + 31T^{2} \)
37 \( 1 - 1.26T + 37T^{2} \)
41 \( 1 + 8.40T + 41T^{2} \)
43 \( 1 + 0.438T + 43T^{2} \)
47 \( 1 - 7.89T + 47T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 + 3.37T + 59T^{2} \)
61 \( 1 + 9.85T + 61T^{2} \)
67 \( 1 - 6.77T + 67T^{2} \)
71 \( 1 - 14.8T + 71T^{2} \)
73 \( 1 + 1.64T + 73T^{2} \)
79 \( 1 - 2.37T + 79T^{2} \)
83 \( 1 + 0.683T + 83T^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 - 3.79T + 97T^{2} \)
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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

−7.80610487119845133485696466136, −6.86618844128654184672590916887, −6.62214076469832431517351941390, −5.79148710433968218709666949865, −5.37004770392155836105711525504, −4.34726947278900970834959717336, −3.45529188660064884997837095399, −2.69616056807863146645319961580, −1.21323140534805997614673559008, −0.50079929770644138123710232522, 0.50079929770644138123710232522, 1.21323140534805997614673559008, 2.69616056807863146645319961580, 3.45529188660064884997837095399, 4.34726947278900970834959717336, 5.37004770392155836105711525504, 5.79148710433968218709666949865, 6.62214076469832431517351941390, 6.86618844128654184672590916887, 7.80610487119845133485696466136

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