Properties

Label 2-8024-1.1-c1-0-192
Degree $2$
Conductor $8024$
Sign $-1$
Analytic cond. $64.0719$
Root an. cond. $8.00449$
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  + 2.56·3-s − 2.56·5-s + 0.561·7-s + 3.56·9-s − 3.12·11-s − 2·13-s − 6.56·15-s + 17-s + 2.56·19-s + 1.43·21-s − 1.12·23-s + 1.56·25-s + 1.43·27-s + 6.56·29-s + 10.2·31-s − 8·33-s − 1.43·35-s − 2·37-s − 5.12·39-s − 3.43·41-s − 8·43-s − 9.12·45-s − 5.12·47-s − 6.68·49-s + 2.56·51-s + 0.561·53-s + 8·55-s + ⋯
L(s)  = 1  + 1.47·3-s − 1.14·5-s + 0.212·7-s + 1.18·9-s − 0.941·11-s − 0.554·13-s − 1.69·15-s + 0.242·17-s + 0.587·19-s + 0.313·21-s − 0.234·23-s + 0.312·25-s + 0.276·27-s + 1.21·29-s + 1.84·31-s − 1.39·33-s − 0.243·35-s − 0.328·37-s − 0.820·39-s − 0.536·41-s − 1.21·43-s − 1.35·45-s − 0.747·47-s − 0.954·49-s + 0.358·51-s + 0.0771·53-s + 1.07·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8024\)    =    \(2^{3} \cdot 17 \cdot 59\)
Sign: $-1$
Analytic conductor: \(64.0719\)
Root analytic conductor: \(8.00449\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8024,\ (\ :1/2),\ -1)\)

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 \( 1 \)
17 \( 1 - T \)
59 \( 1 + T \)
good3 \( 1 - 2.56T + 3T^{2} \)
5 \( 1 + 2.56T + 5T^{2} \)
7 \( 1 - 0.561T + 7T^{2} \)
11 \( 1 + 3.12T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
19 \( 1 - 2.56T + 19T^{2} \)
23 \( 1 + 1.12T + 23T^{2} \)
29 \( 1 - 6.56T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 3.43T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 5.12T + 47T^{2} \)
53 \( 1 - 0.561T + 53T^{2} \)
61 \( 1 + 7.12T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 9.36T + 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 + 0.561T + 79T^{2} \)
83 \( 1 - 6.87T + 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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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

−7.80738419845090117104302167113, −7.12692911406945995050820689645, −6.28810607645959460613950538951, −5.04067500901734614046889511784, −4.61340097443610130069245139158, −3.69443303508238493960751834237, −3.04903918428058675701340130318, −2.55387407158307993426290374595, −1.41633302698794609502255218889, 0, 1.41633302698794609502255218889, 2.55387407158307993426290374595, 3.04903918428058675701340130318, 3.69443303508238493960751834237, 4.61340097443610130069245139158, 5.04067500901734614046889511784, 6.28810607645959460613950538951, 7.12692911406945995050820689645, 7.80738419845090117104302167113

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