Properties

Label 2-8007-1.1-c1-0-165
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  + 0.734·2-s − 3-s − 1.46·4-s − 1.18·5-s − 0.734·6-s − 4.22·7-s − 2.54·8-s + 9-s − 0.873·10-s − 0.131·11-s + 1.46·12-s − 4.92·13-s − 3.10·14-s + 1.18·15-s + 1.05·16-s + 17-s + 0.734·18-s + 4.85·19-s + 1.73·20-s + 4.22·21-s − 0.0964·22-s − 2.00·23-s + 2.54·24-s − 3.58·25-s − 3.61·26-s − 27-s + 6.17·28-s + ⋯
L(s)  = 1  + 0.519·2-s − 0.577·3-s − 0.730·4-s − 0.532·5-s − 0.299·6-s − 1.59·7-s − 0.898·8-s + 0.333·9-s − 0.276·10-s − 0.0396·11-s + 0.421·12-s − 1.36·13-s − 0.828·14-s + 0.307·15-s + 0.264·16-s + 0.242·17-s + 0.173·18-s + 1.11·19-s + 0.388·20-s + 0.921·21-s − 0.0205·22-s − 0.418·23-s + 0.518·24-s − 0.716·25-s − 0.709·26-s − 0.192·27-s + 1.16·28-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 \( 1 - 0.734T + 2T^{2} \)
5 \( 1 + 1.18T + 5T^{2} \)
7 \( 1 + 4.22T + 7T^{2} \)
11 \( 1 + 0.131T + 11T^{2} \)
13 \( 1 + 4.92T + 13T^{2} \)
19 \( 1 - 4.85T + 19T^{2} \)
23 \( 1 + 2.00T + 23T^{2} \)
29 \( 1 - 5.47T + 29T^{2} \)
31 \( 1 - 3.32T + 31T^{2} \)
37 \( 1 - 1.12T + 37T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 - 6.43T + 43T^{2} \)
47 \( 1 - 6.75T + 47T^{2} \)
53 \( 1 + 0.420T + 53T^{2} \)
59 \( 1 + 12.1T + 59T^{2} \)
61 \( 1 - 7.28T + 61T^{2} \)
67 \( 1 + 0.308T + 67T^{2} \)
71 \( 1 + 9.97T + 71T^{2} \)
73 \( 1 + 4.35T + 73T^{2} \)
79 \( 1 + 2.03T + 79T^{2} \)
83 \( 1 + 2.37T + 83T^{2} \)
89 \( 1 + 5.71T + 89T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−7.41563211778069091504071052583, −6.70669461388389847145474952954, −5.86282222521922481375883713609, −5.54445531729257472586583009728, −4.48664725740433557071106888819, −4.13225607342689783006451393410, −3.14945232162878866390268393040, −2.65985628962998597490637031594, −0.824795998322083426342203489997, 0, 0.824795998322083426342203489997, 2.65985628962998597490637031594, 3.14945232162878866390268393040, 4.13225607342689783006451393410, 4.48664725740433557071106888819, 5.54445531729257472586583009728, 5.86282222521922481375883713609, 6.70669461388389847145474952954, 7.41563211778069091504071052583

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