Properties

Label 2-8024-1.1-c1-0-57
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.477·3-s + 1.25·5-s + 1.96·7-s − 2.77·9-s − 5.00·11-s − 3.48·13-s + 0.600·15-s − 17-s − 0.386·19-s + 0.937·21-s + 4.65·23-s − 3.41·25-s − 2.75·27-s + 5.68·29-s − 5.70·31-s − 2.38·33-s + 2.47·35-s + 6.54·37-s − 1.66·39-s + 5.11·41-s + 3.18·43-s − 3.48·45-s + 3.38·47-s − 3.14·49-s − 0.477·51-s + 1.38·53-s − 6.30·55-s + ⋯
L(s)  = 1  + 0.275·3-s + 0.562·5-s + 0.742·7-s − 0.924·9-s − 1.50·11-s − 0.965·13-s + 0.155·15-s − 0.242·17-s − 0.0887·19-s + 0.204·21-s + 0.970·23-s − 0.683·25-s − 0.530·27-s + 1.05·29-s − 1.02·31-s − 0.415·33-s + 0.417·35-s + 1.07·37-s − 0.266·39-s + 0.798·41-s + 0.485·43-s − 0.520·45-s + 0.493·47-s − 0.448·49-s − 0.0668·51-s + 0.190·53-s − 0.849·55-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: \(0\)
Selberg data: \((2,\ 8024,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.903254329\)
\(L(\frac12)\) \(\approx\) \(1.903254329\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.477T + 3T^{2} \)
5 \( 1 - 1.25T + 5T^{2} \)
7 \( 1 - 1.96T + 7T^{2} \)
11 \( 1 + 5.00T + 11T^{2} \)
13 \( 1 + 3.48T + 13T^{2} \)
19 \( 1 + 0.386T + 19T^{2} \)
23 \( 1 - 4.65T + 23T^{2} \)
29 \( 1 - 5.68T + 29T^{2} \)
31 \( 1 + 5.70T + 31T^{2} \)
37 \( 1 - 6.54T + 37T^{2} \)
41 \( 1 - 5.11T + 41T^{2} \)
43 \( 1 - 3.18T + 43T^{2} \)
47 \( 1 - 3.38T + 47T^{2} \)
53 \( 1 - 1.38T + 53T^{2} \)
61 \( 1 + 4.68T + 61T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 - 4.66T + 71T^{2} \)
73 \( 1 - 13.3T + 73T^{2} \)
79 \( 1 - 4.30T + 79T^{2} \)
83 \( 1 + 5.58T + 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 + 0.411T + 97T^{2} \)
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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

−7.85314358258373662517410590072, −7.36807362354124987284206423776, −6.37613639508439888794235547001, −5.58014395725753551079931780593, −5.13402458488368617774885330944, −4.49043863871351418900667346597, −3.30336674040359645618361853705, −2.45899324417741197168227177192, −2.13999219631001868515394067719, −0.63940512587504471671668263133, 0.63940512587504471671668263133, 2.13999219631001868515394067719, 2.45899324417741197168227177192, 3.30336674040359645618361853705, 4.49043863871351418900667346597, 5.13402458488368617774885330944, 5.58014395725753551079931780593, 6.37613639508439888794235547001, 7.36807362354124987284206423776, 7.85314358258373662517410590072

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