Properties

Label 2-8007-1.1-c1-0-401
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  + 2.24·2-s − 3-s + 3.02·4-s + 2.80·5-s − 2.24·6-s − 1.91·7-s + 2.29·8-s + 9-s + 6.28·10-s + 0.785·11-s − 3.02·12-s − 6.00·13-s − 4.30·14-s − 2.80·15-s − 0.898·16-s + 17-s + 2.24·18-s + 3.83·19-s + 8.47·20-s + 1.91·21-s + 1.76·22-s − 5.66·23-s − 2.29·24-s + 2.85·25-s − 13.4·26-s − 27-s − 5.80·28-s + ⋯
L(s)  = 1  + 1.58·2-s − 0.577·3-s + 1.51·4-s + 1.25·5-s − 0.915·6-s − 0.725·7-s + 0.812·8-s + 0.333·9-s + 1.98·10-s + 0.236·11-s − 0.873·12-s − 1.66·13-s − 1.15·14-s − 0.723·15-s − 0.224·16-s + 0.242·17-s + 0.528·18-s + 0.878·19-s + 1.89·20-s + 0.418·21-s + 0.375·22-s − 1.18·23-s − 0.469·24-s + 0.571·25-s − 2.63·26-s − 0.192·27-s − 1.09·28-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 \( 1 - 2.24T + 2T^{2} \)
5 \( 1 - 2.80T + 5T^{2} \)
7 \( 1 + 1.91T + 7T^{2} \)
11 \( 1 - 0.785T + 11T^{2} \)
13 \( 1 + 6.00T + 13T^{2} \)
19 \( 1 - 3.83T + 19T^{2} \)
23 \( 1 + 5.66T + 23T^{2} \)
29 \( 1 + 4.03T + 29T^{2} \)
31 \( 1 + 8.03T + 31T^{2} \)
37 \( 1 - 2.70T + 37T^{2} \)
41 \( 1 + 2.83T + 41T^{2} \)
43 \( 1 + 2.91T + 43T^{2} \)
47 \( 1 - 3.74T + 47T^{2} \)
53 \( 1 + 7.34T + 53T^{2} \)
59 \( 1 + 0.402T + 59T^{2} \)
61 \( 1 - 1.15T + 61T^{2} \)
67 \( 1 + 6.92T + 67T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 + 6.75T + 73T^{2} \)
79 \( 1 - 6.24T + 79T^{2} \)
83 \( 1 - 4.64T + 83T^{2} \)
89 \( 1 + 1.52T + 89T^{2} \)
97 \( 1 + 7.11T + 97T^{2} \)
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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

−7.13314222797786756007275171865, −6.44361712082331755285145551767, −5.93022121143092678674308579364, −5.34317313111492450362730789126, −4.93951754228581535542570113114, −3.97943964554178651333420330932, −3.24714247872711710812141047626, −2.38136210675962988842499675846, −1.73153092499978109344929895438, 0, 1.73153092499978109344929895438, 2.38136210675962988842499675846, 3.24714247872711710812141047626, 3.97943964554178651333420330932, 4.93951754228581535542570113114, 5.34317313111492450362730789126, 5.93022121143092678674308579364, 6.44361712082331755285145551767, 7.13314222797786756007275171865

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