Properties

Label 2-8007-1.1-c1-0-286
Degree $2$
Conductor $8007$
Sign $-1$
Analytic cond. $63.9362$
Root an. cond. $7.99601$
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  − 2.11·2-s − 3-s + 2.48·4-s + 2.70·5-s + 2.11·6-s + 2.58·7-s − 1.01·8-s + 9-s − 5.72·10-s − 4.73·11-s − 2.48·12-s + 0.962·13-s − 5.47·14-s − 2.70·15-s − 2.80·16-s + 17-s − 2.11·18-s − 1.82·19-s + 6.70·20-s − 2.58·21-s + 10.0·22-s − 1.57·23-s + 1.01·24-s + 2.30·25-s − 2.03·26-s − 27-s + 6.41·28-s + ⋯
L(s)  = 1  − 1.49·2-s − 0.577·3-s + 1.24·4-s + 1.20·5-s + 0.864·6-s + 0.977·7-s − 0.359·8-s + 0.333·9-s − 1.80·10-s − 1.42·11-s − 0.716·12-s + 0.266·13-s − 1.46·14-s − 0.697·15-s − 0.701·16-s + 0.242·17-s − 0.498·18-s − 0.419·19-s + 1.49·20-s − 0.564·21-s + 2.13·22-s − 0.328·23-s + 0.207·24-s + 0.460·25-s − 0.399·26-s − 0.192·27-s + 1.21·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8007\)    =    \(3 \cdot 17 \cdot 157\)
Sign: $-1$
Analytic conductor: \(63.9362\)
Root analytic conductor: \(7.99601\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8007,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
17 \( 1 - T \)
157 \( 1 - T \)
good2 \( 1 + 2.11T + 2T^{2} \)
5 \( 1 - 2.70T + 5T^{2} \)
7 \( 1 - 2.58T + 7T^{2} \)
11 \( 1 + 4.73T + 11T^{2} \)
13 \( 1 - 0.962T + 13T^{2} \)
19 \( 1 + 1.82T + 19T^{2} \)
23 \( 1 + 1.57T + 23T^{2} \)
29 \( 1 + 3.37T + 29T^{2} \)
31 \( 1 + 7.71T + 31T^{2} \)
37 \( 1 + 4.06T + 37T^{2} \)
41 \( 1 - 2.50T + 41T^{2} \)
43 \( 1 - 5.29T + 43T^{2} \)
47 \( 1 - 9.84T + 47T^{2} \)
53 \( 1 + 2.01T + 53T^{2} \)
59 \( 1 - 13.9T + 59T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 - 1.38T + 71T^{2} \)
73 \( 1 - 6.22T + 73T^{2} \)
79 \( 1 + 10.5T + 79T^{2} \)
83 \( 1 + 0.517T + 83T^{2} \)
89 \( 1 + 1.13T + 89T^{2} \)
97 \( 1 + 5.39T + 97T^{2} \)
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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

−7.59123384386308307155293424395, −7.09193696854521543438976698332, −6.10955124137405273279343069384, −5.49674951705281301580068391024, −5.01588602290538004636652045672, −3.95856299495892766718994287583, −2.45450941625223709815860250907, −1.98423884944989440209421138090, −1.17315480677342175449357608913, 0, 1.17315480677342175449357608913, 1.98423884944989440209421138090, 2.45450941625223709815860250907, 3.95856299495892766718994287583, 5.01588602290538004636652045672, 5.49674951705281301580068391024, 6.10955124137405273279343069384, 7.09193696854521543438976698332, 7.59123384386308307155293424395

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