Properties

Label 2-8043-1.1-c1-0-241
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  − 0.810·2-s + 3-s − 1.34·4-s + 0.143·5-s − 0.810·6-s − 7-s + 2.70·8-s + 9-s − 0.116·10-s − 0.888·11-s − 1.34·12-s − 0.424·13-s + 0.810·14-s + 0.143·15-s + 0.489·16-s − 0.886·17-s − 0.810·18-s − 8.58·19-s − 0.192·20-s − 21-s + 0.720·22-s + 4.66·23-s + 2.70·24-s − 4.97·25-s + 0.343·26-s + 27-s + 1.34·28-s + ⋯
L(s)  = 1  − 0.573·2-s + 0.577·3-s − 0.671·4-s + 0.0641·5-s − 0.330·6-s − 0.377·7-s + 0.958·8-s + 0.333·9-s − 0.0367·10-s − 0.268·11-s − 0.387·12-s − 0.117·13-s + 0.216·14-s + 0.0370·15-s + 0.122·16-s − 0.215·17-s − 0.191·18-s − 1.96·19-s − 0.0430·20-s − 0.218·21-s + 0.153·22-s + 0.972·23-s + 0.553·24-s − 0.995·25-s + 0.0674·26-s + 0.192·27-s + 0.253·28-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 \( 1 + 0.810T + 2T^{2} \)
5 \( 1 - 0.143T + 5T^{2} \)
11 \( 1 + 0.888T + 11T^{2} \)
13 \( 1 + 0.424T + 13T^{2} \)
17 \( 1 + 0.886T + 17T^{2} \)
19 \( 1 + 8.58T + 19T^{2} \)
23 \( 1 - 4.66T + 23T^{2} \)
29 \( 1 - 7.67T + 29T^{2} \)
31 \( 1 - 5.71T + 31T^{2} \)
37 \( 1 + 3.86T + 37T^{2} \)
41 \( 1 - 3.14T + 41T^{2} \)
43 \( 1 - 4.97T + 43T^{2} \)
47 \( 1 - 8.47T + 47T^{2} \)
53 \( 1 - 0.270T + 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 + 14.4T + 61T^{2} \)
67 \( 1 - 8.57T + 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 - 3.15T + 73T^{2} \)
79 \( 1 + 8.70T + 79T^{2} \)
83 \( 1 + 15.0T + 83T^{2} \)
89 \( 1 - 4.00T + 89T^{2} \)
97 \( 1 + 8.01T + 97T^{2} \)
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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

−7.66350628311731383715340063851, −6.94406341733516513615195326557, −6.23633786117511106755277253226, −5.34911573578816955788403347088, −4.35955379156130775507539049783, −4.14638570413755308606637623780, −2.95037129591724616513847189824, −2.24621733638011571774222328372, −1.12958238908830229180383132468, 0, 1.12958238908830229180383132468, 2.24621733638011571774222328372, 2.95037129591724616513847189824, 4.14638570413755308606637623780, 4.35955379156130775507539049783, 5.34911573578816955788403347088, 6.23633786117511106755277253226, 6.94406341733516513615195326557, 7.66350628311731383715340063851

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