Properties

Label 2-8024-1.1-c1-0-82
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  − 0.288·3-s + 2.13·5-s + 1.97·7-s − 2.91·9-s − 1.51·11-s + 1.35·13-s − 0.616·15-s + 17-s + 3.95·19-s − 0.567·21-s − 2.03·23-s − 0.420·25-s + 1.70·27-s − 10.0·29-s − 2.29·31-s + 0.436·33-s + 4.21·35-s + 6.48·37-s − 0.389·39-s + 9.92·41-s − 2.66·43-s − 6.24·45-s + 12.6·47-s − 3.11·49-s − 0.288·51-s + 9.27·53-s − 3.23·55-s + ⋯
L(s)  = 1  − 0.166·3-s + 0.956·5-s + 0.745·7-s − 0.972·9-s − 0.456·11-s + 0.374·13-s − 0.159·15-s + 0.242·17-s + 0.906·19-s − 0.123·21-s − 0.424·23-s − 0.0841·25-s + 0.328·27-s − 1.86·29-s − 0.411·31-s + 0.0759·33-s + 0.712·35-s + 1.06·37-s − 0.0623·39-s + 1.54·41-s − 0.406·43-s − 0.930·45-s + 1.85·47-s − 0.444·49-s − 0.0403·51-s + 1.27·53-s − 0.436·55-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

\(L(1)\) \(\approx\) \(2.322420555\)
\(L(\frac12)\) \(\approx\) \(2.322420555\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.288T + 3T^{2} \)
5 \( 1 - 2.13T + 5T^{2} \)
7 \( 1 - 1.97T + 7T^{2} \)
11 \( 1 + 1.51T + 11T^{2} \)
13 \( 1 - 1.35T + 13T^{2} \)
19 \( 1 - 3.95T + 19T^{2} \)
23 \( 1 + 2.03T + 23T^{2} \)
29 \( 1 + 10.0T + 29T^{2} \)
31 \( 1 + 2.29T + 31T^{2} \)
37 \( 1 - 6.48T + 37T^{2} \)
41 \( 1 - 9.92T + 41T^{2} \)
43 \( 1 + 2.66T + 43T^{2} \)
47 \( 1 - 12.6T + 47T^{2} \)
53 \( 1 - 9.27T + 53T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 + 13.8T + 67T^{2} \)
71 \( 1 - 3.17T + 71T^{2} \)
73 \( 1 + 5.27T + 73T^{2} \)
79 \( 1 + 7.76T + 79T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 - 1.97T + 89T^{2} \)
97 \( 1 - 6.39T + 97T^{2} \)
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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

−7.70920378929476187650045126915, −7.35416347975422449338505281207, −6.12784801606086017521618771354, −5.68385796990081501137429137496, −5.36472463544220407552370361915, −4.35324419253639613667597382835, −3.46259358212479758148199104850, −2.50245840334697140747824322180, −1.87673259958868079572798615594, −0.76138268875345036390991182594, 0.76138268875345036390991182594, 1.87673259958868079572798615594, 2.50245840334697140747824322180, 3.46259358212479758148199104850, 4.35324419253639613667597382835, 5.36472463544220407552370361915, 5.68385796990081501137429137496, 6.12784801606086017521618771354, 7.35416347975422449338505281207, 7.70920378929476187650045126915

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