Properties

Label 2-8007-1.1-c1-0-256
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  + 0.365·2-s + 3-s − 1.86·4-s − 3.97·5-s + 0.365·6-s + 3.07·7-s − 1.41·8-s + 9-s − 1.45·10-s + 0.220·11-s − 1.86·12-s − 3.98·13-s + 1.12·14-s − 3.97·15-s + 3.21·16-s + 17-s + 0.365·18-s − 0.328·19-s + 7.42·20-s + 3.07·21-s + 0.0804·22-s − 2.08·23-s − 1.41·24-s + 10.8·25-s − 1.45·26-s + 27-s − 5.74·28-s + ⋯
L(s)  = 1  + 0.258·2-s + 0.577·3-s − 0.933·4-s − 1.77·5-s + 0.149·6-s + 1.16·7-s − 0.499·8-s + 0.333·9-s − 0.460·10-s + 0.0663·11-s − 0.538·12-s − 1.10·13-s + 0.300·14-s − 1.02·15-s + 0.803·16-s + 0.242·17-s + 0.0861·18-s − 0.0753·19-s + 1.66·20-s + 0.671·21-s + 0.0171·22-s − 0.433·23-s − 0.288·24-s + 2.16·25-s − 0.285·26-s + 0.192·27-s − 1.08·28-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 \( 1 - 0.365T + 2T^{2} \)
5 \( 1 + 3.97T + 5T^{2} \)
7 \( 1 - 3.07T + 7T^{2} \)
11 \( 1 - 0.220T + 11T^{2} \)
13 \( 1 + 3.98T + 13T^{2} \)
19 \( 1 + 0.328T + 19T^{2} \)
23 \( 1 + 2.08T + 23T^{2} \)
29 \( 1 + 1.31T + 29T^{2} \)
31 \( 1 + 0.177T + 31T^{2} \)
37 \( 1 - 8.74T + 37T^{2} \)
41 \( 1 + 1.77T + 41T^{2} \)
43 \( 1 - 7.05T + 43T^{2} \)
47 \( 1 + 3.10T + 47T^{2} \)
53 \( 1 + 7.51T + 53T^{2} \)
59 \( 1 - 1.69T + 59T^{2} \)
61 \( 1 - 6.89T + 61T^{2} \)
67 \( 1 + 2.47T + 67T^{2} \)
71 \( 1 + 0.954T + 71T^{2} \)
73 \( 1 + 7.59T + 73T^{2} \)
79 \( 1 - 1.89T + 79T^{2} \)
83 \( 1 - 1.21T + 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 - 6.57T + 97T^{2} \)
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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

−7.68568365334936275621550851859, −7.20362345140698948454840752205, −5.99734774730694014562543745823, −4.91076014781893933866922705726, −4.64461658789017407881745791022, −3.99669751799542938008999158527, −3.35580589721727613525794933158, −2.42458639839715162500143857584, −1.09491862335192492952708883871, 0, 1.09491862335192492952708883871, 2.42458639839715162500143857584, 3.35580589721727613525794933158, 3.99669751799542938008999158527, 4.64461658789017407881745791022, 4.91076014781893933866922705726, 5.99734774730694014562543745823, 7.20362345140698948454840752205, 7.68568365334936275621550851859

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