Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 157 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.751·2-s + 3-s − 1.43·4-s + 3.06·5-s + 0.751·6-s − 2.02·7-s − 2.58·8-s + 9-s + 2.30·10-s − 2.87·11-s − 1.43·12-s − 0.853·13-s − 1.51·14-s + 3.06·15-s + 0.928·16-s − 17-s + 0.751·18-s + 1.01·19-s − 4.39·20-s − 2.02·21-s − 2.16·22-s + 2.89·23-s − 2.58·24-s + 4.36·25-s − 0.641·26-s + 27-s + 2.89·28-s + ⋯
L(s)  = 1  + 0.531·2-s + 0.577·3-s − 0.717·4-s + 1.36·5-s + 0.306·6-s − 0.763·7-s − 0.912·8-s + 0.333·9-s + 0.727·10-s − 0.866·11-s − 0.414·12-s − 0.236·13-s − 0.406·14-s + 0.790·15-s + 0.232·16-s − 0.242·17-s + 0.177·18-s + 0.233·19-s − 0.981·20-s − 0.440·21-s − 0.460·22-s + 0.604·23-s − 0.527·24-s + 0.873·25-s − 0.125·26-s + 0.192·27-s + 0.547·28-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

\( d \)  =  \(2\)
\( N \)  =  \(8007\)    =    \(3 \cdot 17 \cdot 157\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8007} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8007,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;17,\;157\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17,\;157\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
17 \( 1 + T \)
157 \( 1 - T \)
good2 \( 1 - 0.751T + 2T^{2} \)
5 \( 1 - 3.06T + 5T^{2} \)
7 \( 1 + 2.02T + 7T^{2} \)
11 \( 1 + 2.87T + 11T^{2} \)
13 \( 1 + 0.853T + 13T^{2} \)
19 \( 1 - 1.01T + 19T^{2} \)
23 \( 1 - 2.89T + 23T^{2} \)
29 \( 1 + 7.11T + 29T^{2} \)
31 \( 1 - 1.50T + 31T^{2} \)
37 \( 1 - 8.80T + 37T^{2} \)
41 \( 1 - 4.79T + 41T^{2} \)
43 \( 1 + 0.143T + 43T^{2} \)
47 \( 1 + 6.98T + 47T^{2} \)
53 \( 1 + 7.81T + 53T^{2} \)
59 \( 1 + 9.82T + 59T^{2} \)
61 \( 1 + 8.44T + 61T^{2} \)
67 \( 1 + 0.210T + 67T^{2} \)
71 \( 1 + 4.66T + 71T^{2} \)
73 \( 1 - 4.89T + 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 - 2.85T + 83T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.59589227428451644009611084953, −6.50088686213677804988086665856, −6.08076703299366673415518257791, −5.31141097728644968584581208173, −4.79097989552600214402259356437, −3.87027419914777700075283959809, −2.97586600389741487276287735235, −2.56064098297597814187854258236, −1.43736950349119557434044267329, 0, 1.43736950349119557434044267329, 2.56064098297597814187854258236, 2.97586600389741487276287735235, 3.87027419914777700075283959809, 4.79097989552600214402259356437, 5.31141097728644968584581208173, 6.08076703299366673415518257791, 6.50088686213677804988086665856, 7.59589227428451644009611084953

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