Properties

Label 2-8007-1.1-c1-0-231
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.68·2-s − 3-s + 5.19·4-s − 0.525·5-s + 2.68·6-s + 1.96·7-s − 8.58·8-s + 9-s + 1.40·10-s + 4.22·11-s − 5.19·12-s − 5.20·13-s − 5.28·14-s + 0.525·15-s + 12.6·16-s + 17-s − 2.68·18-s − 2.46·19-s − 2.73·20-s − 1.96·21-s − 11.3·22-s − 3.47·23-s + 8.58·24-s − 4.72·25-s + 13.9·26-s − 27-s + 10.2·28-s + ⋯
L(s)  = 1  − 1.89·2-s − 0.577·3-s + 2.59·4-s − 0.234·5-s + 1.09·6-s + 0.744·7-s − 3.03·8-s + 0.333·9-s + 0.445·10-s + 1.27·11-s − 1.50·12-s − 1.44·13-s − 1.41·14-s + 0.135·15-s + 3.15·16-s + 0.242·17-s − 0.632·18-s − 0.565·19-s − 0.610·20-s − 0.429·21-s − 2.41·22-s − 0.725·23-s + 1.75·24-s − 0.944·25-s + 2.73·26-s − 0.192·27-s + 1.93·28-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 \( 1 + 2.68T + 2T^{2} \)
5 \( 1 + 0.525T + 5T^{2} \)
7 \( 1 - 1.96T + 7T^{2} \)
11 \( 1 - 4.22T + 11T^{2} \)
13 \( 1 + 5.20T + 13T^{2} \)
19 \( 1 + 2.46T + 19T^{2} \)
23 \( 1 + 3.47T + 23T^{2} \)
29 \( 1 - 6.01T + 29T^{2} \)
31 \( 1 + 1.06T + 31T^{2} \)
37 \( 1 + 2.45T + 37T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 + 1.10T + 43T^{2} \)
47 \( 1 - 3.91T + 47T^{2} \)
53 \( 1 + 0.313T + 53T^{2} \)
59 \( 1 - 9.87T + 59T^{2} \)
61 \( 1 + 1.73T + 61T^{2} \)
67 \( 1 + 0.944T + 67T^{2} \)
71 \( 1 + 4.49T + 71T^{2} \)
73 \( 1 + 5.99T + 73T^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 + 14.1T + 83T^{2} \)
89 \( 1 + 2.39T + 89T^{2} \)
97 \( 1 + 3.48T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.59718675526930414426682470577, −7.07380726749721119528540486013, −6.38549065007413835658923267887, −5.75217123221312351378999694018, −4.70181752998404770793303785497, −3.87272964422758633180750210427, −2.58243813190312615312378068769, −1.86245433470806011502316294610, −1.02942543226846758077244451635, 0, 1.02942543226846758077244451635, 1.86245433470806011502316294610, 2.58243813190312615312378068769, 3.87272964422758633180750210427, 4.70181752998404770793303785497, 5.75217123221312351378999694018, 6.38549065007413835658923267887, 7.07380726749721119528540486013, 7.59718675526930414426682470577

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