Properties

Label 2-8007-1.1-c1-0-380
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  + 1.51·2-s + 3-s + 0.283·4-s − 0.836·5-s + 1.51·6-s + 4.41·7-s − 2.59·8-s + 9-s − 1.26·10-s − 3.77·11-s + 0.283·12-s − 5.83·13-s + 6.66·14-s − 0.836·15-s − 4.48·16-s − 17-s + 1.51·18-s + 4.09·19-s − 0.236·20-s + 4.41·21-s − 5.70·22-s + 9.55·23-s − 2.59·24-s − 4.30·25-s − 8.82·26-s + 27-s + 1.24·28-s + ⋯
L(s)  = 1  + 1.06·2-s + 0.577·3-s + 0.141·4-s − 0.373·5-s + 0.616·6-s + 1.66·7-s − 0.917·8-s + 0.333·9-s − 0.399·10-s − 1.13·11-s + 0.0816·12-s − 1.61·13-s + 1.78·14-s − 0.215·15-s − 1.12·16-s − 0.242·17-s + 0.356·18-s + 0.939·19-s − 0.0529·20-s + 0.962·21-s − 1.21·22-s + 1.99·23-s − 0.529·24-s − 0.860·25-s − 1.73·26-s + 0.192·27-s + 0.235·28-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 \( 1 - 1.51T + 2T^{2} \)
5 \( 1 + 0.836T + 5T^{2} \)
7 \( 1 - 4.41T + 7T^{2} \)
11 \( 1 + 3.77T + 11T^{2} \)
13 \( 1 + 5.83T + 13T^{2} \)
19 \( 1 - 4.09T + 19T^{2} \)
23 \( 1 - 9.55T + 23T^{2} \)
29 \( 1 + 9.47T + 29T^{2} \)
31 \( 1 - 0.516T + 31T^{2} \)
37 \( 1 - 1.47T + 37T^{2} \)
41 \( 1 + 2.87T + 41T^{2} \)
43 \( 1 - 3.76T + 43T^{2} \)
47 \( 1 + 12.0T + 47T^{2} \)
53 \( 1 - 1.99T + 53T^{2} \)
59 \( 1 - 3.16T + 59T^{2} \)
61 \( 1 + 6.64T + 61T^{2} \)
67 \( 1 + 2.31T + 67T^{2} \)
71 \( 1 + 1.24T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 + 5.66T + 79T^{2} \)
83 \( 1 + 7.20T + 83T^{2} \)
89 \( 1 + 16.5T + 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−7.49061832216513408491753083009, −7.04395290698845611221240852358, −5.61343564385124068973938978879, −5.18830741019234897558429953184, −4.73851827034418491457647713921, −4.10224779237025297305352251913, −3.07628632073417436689626157044, −2.55857199675575979606460891621, −1.58131104531063412740141268616, 0, 1.58131104531063412740141268616, 2.55857199675575979606460891621, 3.07628632073417436689626157044, 4.10224779237025297305352251913, 4.73851827034418491457647713921, 5.18830741019234897558429953184, 5.61343564385124068973938978879, 7.04395290698845611221240852358, 7.49061832216513408491753083009

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