Properties

Label 2-8024-1.1-c1-0-112
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  + 1.91·3-s + 3.63·5-s − 4.69·7-s + 0.651·9-s + 3.01·11-s + 5.45·13-s + 6.94·15-s + 17-s − 2.51·19-s − 8.96·21-s − 2.25·23-s + 8.20·25-s − 4.48·27-s + 9.37·29-s + 6.90·31-s + 5.75·33-s − 17.0·35-s + 0.180·37-s + 10.4·39-s − 0.746·41-s − 6.45·43-s + 2.36·45-s − 11.6·47-s + 15.0·49-s + 1.91·51-s − 1.10·53-s + 10.9·55-s + ⋯
L(s)  = 1  + 1.10·3-s + 1.62·5-s − 1.77·7-s + 0.217·9-s + 0.907·11-s + 1.51·13-s + 1.79·15-s + 0.242·17-s − 0.576·19-s − 1.95·21-s − 0.471·23-s + 1.64·25-s − 0.863·27-s + 1.74·29-s + 1.23·31-s + 1.00·33-s − 2.88·35-s + 0.0297·37-s + 1.66·39-s − 0.116·41-s − 0.983·43-s + 0.353·45-s − 1.69·47-s + 2.14·49-s + 0.267·51-s − 0.151·53-s + 1.47·55-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

\(L(1)\) \(\approx\) \(4.015063959\)
\(L(\frac12)\) \(\approx\) \(4.015063959\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.91T + 3T^{2} \)
5 \( 1 - 3.63T + 5T^{2} \)
7 \( 1 + 4.69T + 7T^{2} \)
11 \( 1 - 3.01T + 11T^{2} \)
13 \( 1 - 5.45T + 13T^{2} \)
19 \( 1 + 2.51T + 19T^{2} \)
23 \( 1 + 2.25T + 23T^{2} \)
29 \( 1 - 9.37T + 29T^{2} \)
31 \( 1 - 6.90T + 31T^{2} \)
37 \( 1 - 0.180T + 37T^{2} \)
41 \( 1 + 0.746T + 41T^{2} \)
43 \( 1 + 6.45T + 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 + 1.10T + 53T^{2} \)
61 \( 1 - 0.347T + 61T^{2} \)
67 \( 1 - 14.9T + 67T^{2} \)
71 \( 1 - 9.72T + 71T^{2} \)
73 \( 1 - 12.5T + 73T^{2} \)
79 \( 1 - 7.14T + 79T^{2} \)
83 \( 1 - 1.12T + 83T^{2} \)
89 \( 1 - 2.80T + 89T^{2} \)
97 \( 1 + 4.73T + 97T^{2} \)
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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

−8.290587857262538835707299132405, −6.70212999680540107113554443062, −6.44953852522068756487027372326, −6.13964040462893190563585611301, −5.14708011341206729058480051936, −3.92988657537777471317571223015, −3.36526525853061322293121343659, −2.73494377409663472211544081923, −1.95427672117851446141375614908, −0.968055336734428193033877081776, 0.968055336734428193033877081776, 1.95427672117851446141375614908, 2.73494377409663472211544081923, 3.36526525853061322293121343659, 3.92988657537777471317571223015, 5.14708011341206729058480051936, 6.13964040462893190563585611301, 6.44953852522068756487027372326, 6.70212999680540107113554443062, 8.290587857262538835707299132405

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