Properties

Label 2-8007-1.1-c1-0-390
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  + 2.42·2-s + 3-s + 3.86·4-s − 4.19·5-s + 2.42·6-s − 0.311·7-s + 4.51·8-s + 9-s − 10.1·10-s + 1.14·11-s + 3.86·12-s − 0.0687·13-s − 0.754·14-s − 4.19·15-s + 3.20·16-s + 17-s + 2.42·18-s − 3.46·19-s − 16.2·20-s − 0.311·21-s + 2.78·22-s − 6.85·23-s + 4.51·24-s + 12.5·25-s − 0.166·26-s + 27-s − 1.20·28-s + ⋯
L(s)  = 1  + 1.71·2-s + 0.577·3-s + 1.93·4-s − 1.87·5-s + 0.988·6-s − 0.117·7-s + 1.59·8-s + 0.333·9-s − 3.21·10-s + 0.346·11-s + 1.11·12-s − 0.0190·13-s − 0.201·14-s − 1.08·15-s + 0.800·16-s + 0.242·17-s + 0.570·18-s − 0.795·19-s − 3.62·20-s − 0.0679·21-s + 0.593·22-s − 1.42·23-s + 0.921·24-s + 2.51·25-s − 0.0326·26-s + 0.192·27-s − 0.227·28-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 \( 1 - 2.42T + 2T^{2} \)
5 \( 1 + 4.19T + 5T^{2} \)
7 \( 1 + 0.311T + 7T^{2} \)
11 \( 1 - 1.14T + 11T^{2} \)
13 \( 1 + 0.0687T + 13T^{2} \)
19 \( 1 + 3.46T + 19T^{2} \)
23 \( 1 + 6.85T + 23T^{2} \)
29 \( 1 - 3.49T + 29T^{2} \)
31 \( 1 + 2.04T + 31T^{2} \)
37 \( 1 + 0.351T + 37T^{2} \)
41 \( 1 - 6.21T + 41T^{2} \)
43 \( 1 + 11.8T + 43T^{2} \)
47 \( 1 + 1.87T + 47T^{2} \)
53 \( 1 + 1.70T + 53T^{2} \)
59 \( 1 + 1.76T + 59T^{2} \)
61 \( 1 + 2.33T + 61T^{2} \)
67 \( 1 + 9.21T + 67T^{2} \)
71 \( 1 + 3.55T + 71T^{2} \)
73 \( 1 - 3.46T + 73T^{2} \)
79 \( 1 - 7.20T + 79T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 - 2.17T + 97T^{2} \)
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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

−7.33999699265172095789777580063, −6.72250959352983208736985853145, −6.11498614355061912369487720527, −5.07461419087478756097654000771, −4.35612827913518859890724532546, −4.01985787327720899235971117403, −3.35559479866607465295453049343, −2.77864077553709742213793735871, −1.65122103842030231688995674902, 0, 1.65122103842030231688995674902, 2.77864077553709742213793735871, 3.35559479866607465295453049343, 4.01985787327720899235971117403, 4.35612827913518859890724532546, 5.07461419087478756097654000771, 6.11498614355061912369487720527, 6.72250959352983208736985853145, 7.33999699265172095789777580063

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