Properties

Label 2-8043-1.1-c1-0-290
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  − 2.68·2-s − 3-s + 5.21·4-s + 2.43·5-s + 2.68·6-s − 7-s − 8.62·8-s + 9-s − 6.53·10-s − 0.147·11-s − 5.21·12-s + 3.06·13-s + 2.68·14-s − 2.43·15-s + 12.7·16-s + 3.88·17-s − 2.68·18-s + 2.57·19-s + 12.6·20-s + 21-s + 0.395·22-s + 4.81·23-s + 8.62·24-s + 0.922·25-s − 8.22·26-s − 27-s − 5.21·28-s + ⋯
L(s)  = 1  − 1.89·2-s − 0.577·3-s + 2.60·4-s + 1.08·5-s + 1.09·6-s − 0.377·7-s − 3.04·8-s + 0.333·9-s − 2.06·10-s − 0.0443·11-s − 1.50·12-s + 0.849·13-s + 0.717·14-s − 0.628·15-s + 3.18·16-s + 0.943·17-s − 0.632·18-s + 0.589·19-s + 2.83·20-s + 0.218·21-s + 0.0842·22-s + 1.00·23-s + 1.76·24-s + 0.184·25-s − 1.61·26-s − 0.192·27-s − 0.984·28-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 \( 1 + 2.68T + 2T^{2} \)
5 \( 1 - 2.43T + 5T^{2} \)
11 \( 1 + 0.147T + 11T^{2} \)
13 \( 1 - 3.06T + 13T^{2} \)
17 \( 1 - 3.88T + 17T^{2} \)
19 \( 1 - 2.57T + 19T^{2} \)
23 \( 1 - 4.81T + 23T^{2} \)
29 \( 1 - 3.26T + 29T^{2} \)
31 \( 1 - 0.103T + 31T^{2} \)
37 \( 1 + 10.5T + 37T^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 + 7.89T + 43T^{2} \)
47 \( 1 - 3.42T + 47T^{2} \)
53 \( 1 + 6.68T + 53T^{2} \)
59 \( 1 - 0.609T + 59T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 + 6.19T + 67T^{2} \)
71 \( 1 + 0.516T + 71T^{2} \)
73 \( 1 + 1.60T + 73T^{2} \)
79 \( 1 + 2.34T + 79T^{2} \)
83 \( 1 + 8.68T + 83T^{2} \)
89 \( 1 + 4.28T + 89T^{2} \)
97 \( 1 + 14.0T + 97T^{2} \)
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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

−7.52060525541786896697192816143, −6.84793718793295870318328796249, −6.38276139490847562221960079655, −5.70264659147715950101663910998, −5.08824134965176104708058726187, −3.45972621555858546379541599932, −2.82472138904873791247097839098, −1.59974695369257422376108725751, −1.28837266621685142774755196471, 0, 1.28837266621685142774755196471, 1.59974695369257422376108725751, 2.82472138904873791247097839098, 3.45972621555858546379541599932, 5.08824134965176104708058726187, 5.70264659147715950101663910998, 6.38276139490847562221960079655, 6.84793718793295870318328796249, 7.52060525541786896697192816143

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