Properties

Label 2-8043-1.1-c1-0-353
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  + 0.609·2-s + 3-s − 1.62·4-s + 3.04·5-s + 0.609·6-s + 7-s − 2.21·8-s + 9-s + 1.85·10-s − 2.44·11-s − 1.62·12-s − 3.77·13-s + 0.609·14-s + 3.04·15-s + 1.90·16-s + 3.61·17-s + 0.609·18-s + 7.07·19-s − 4.95·20-s + 21-s − 1.49·22-s − 6.33·23-s − 2.21·24-s + 4.26·25-s − 2.29·26-s + 27-s − 1.62·28-s + ⋯
L(s)  = 1  + 0.431·2-s + 0.577·3-s − 0.813·4-s + 1.36·5-s + 0.248·6-s + 0.377·7-s − 0.782·8-s + 0.333·9-s + 0.587·10-s − 0.737·11-s − 0.469·12-s − 1.04·13-s + 0.163·14-s + 0.785·15-s + 0.476·16-s + 0.875·17-s + 0.143·18-s + 1.62·19-s − 1.10·20-s + 0.218·21-s − 0.318·22-s − 1.32·23-s − 0.451·24-s + 0.853·25-s − 0.451·26-s + 0.192·27-s − 0.307·28-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 \( 1 - 0.609T + 2T^{2} \)
5 \( 1 - 3.04T + 5T^{2} \)
11 \( 1 + 2.44T + 11T^{2} \)
13 \( 1 + 3.77T + 13T^{2} \)
17 \( 1 - 3.61T + 17T^{2} \)
19 \( 1 - 7.07T + 19T^{2} \)
23 \( 1 + 6.33T + 23T^{2} \)
29 \( 1 + 9.12T + 29T^{2} \)
31 \( 1 + 9.05T + 31T^{2} \)
37 \( 1 + 10.1T + 37T^{2} \)
41 \( 1 + 9.90T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 + 9.22T + 47T^{2} \)
53 \( 1 - 8.85T + 53T^{2} \)
59 \( 1 - 4.48T + 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 - 2.37T + 67T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 - 5.97T + 73T^{2} \)
79 \( 1 - 13.3T + 79T^{2} \)
83 \( 1 + 17.2T + 83T^{2} \)
89 \( 1 - 1.08T + 89T^{2} \)
97 \( 1 - 3.75T + 97T^{2} \)
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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

−7.54207681222791049681792338579, −6.85964960604459917395771560749, −5.63473038181931044551647678107, −5.35049376959258006770244054177, −5.00836899888955903150439512175, −3.68852580695820446295001076588, −3.29057043194399920745185712193, −2.16487335186250396653932805378, −1.60488006354703883681114719195, 0, 1.60488006354703883681114719195, 2.16487335186250396653932805378, 3.29057043194399920745185712193, 3.68852580695820446295001076588, 5.00836899888955903150439512175, 5.35049376959258006770244054177, 5.63473038181931044551647678107, 6.85964960604459917395771560749, 7.54207681222791049681792338579

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