Properties

Label 2-8043-1.1-c1-0-300
Degree $2$
Conductor $8043$
Sign $-1$
Analytic cond. $64.2236$
Root an. cond. $8.01396$
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  + 1.60·2-s + 3-s + 0.568·4-s − 4.39·5-s + 1.60·6-s + 7-s − 2.29·8-s + 9-s − 7.04·10-s + 3.13·11-s + 0.568·12-s + 0.710·13-s + 1.60·14-s − 4.39·15-s − 4.81·16-s − 3.80·17-s + 1.60·18-s + 4.84·19-s − 2.49·20-s + 21-s + 5.02·22-s − 2.77·23-s − 2.29·24-s + 14.3·25-s + 1.13·26-s + 27-s + 0.568·28-s + ⋯
L(s)  = 1  + 1.13·2-s + 0.577·3-s + 0.284·4-s − 1.96·5-s + 0.654·6-s + 0.377·7-s − 0.811·8-s + 0.333·9-s − 2.22·10-s + 0.945·11-s + 0.164·12-s + 0.197·13-s + 0.428·14-s − 1.13·15-s − 1.20·16-s − 0.923·17-s + 0.377·18-s + 1.11·19-s − 0.558·20-s + 0.218·21-s + 1.07·22-s − 0.579·23-s − 0.468·24-s + 2.86·25-s + 0.223·26-s + 0.192·27-s + 0.107·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8043\)    =    \(3 \cdot 7 \cdot 383\)
Sign: $-1$
Analytic conductor: \(64.2236\)
Root analytic conductor: \(8.01396\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8043,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
7 \( 1 - T \)
383 \( 1 + T \)
good2 \( 1 - 1.60T + 2T^{2} \)
5 \( 1 + 4.39T + 5T^{2} \)
11 \( 1 - 3.13T + 11T^{2} \)
13 \( 1 - 0.710T + 13T^{2} \)
17 \( 1 + 3.80T + 17T^{2} \)
19 \( 1 - 4.84T + 19T^{2} \)
23 \( 1 + 2.77T + 23T^{2} \)
29 \( 1 - 1.87T + 29T^{2} \)
31 \( 1 - 8.79T + 31T^{2} \)
37 \( 1 + 11.5T + 37T^{2} \)
41 \( 1 + 3.26T + 41T^{2} \)
43 \( 1 - 3.29T + 43T^{2} \)
47 \( 1 + 6.85T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 - 1.04T + 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 + 12.8T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + 1.27T + 83T^{2} \)
89 \( 1 + 8.72T + 89T^{2} \)
97 \( 1 - 16.8T + 97T^{2} \)
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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

−7.36559925126403021090090655003, −6.85746137042598671849918913575, −6.12065041374371294669275957419, −4.88584273196312754962909350782, −4.60771160252362425050625266362, −3.86165182181422559681805604026, −3.43745062864305136879989945809, −2.73683035613487864615648341456, −1.31433367030452505031348541882, 0, 1.31433367030452505031348541882, 2.73683035613487864615648341456, 3.43745062864305136879989945809, 3.86165182181422559681805604026, 4.60771160252362425050625266362, 4.88584273196312754962909350782, 6.12065041374371294669275957419, 6.85746137042598671849918913575, 7.36559925126403021090090655003

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