Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 157 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.11·2-s − 3-s + 2.47·4-s + 2.21·5-s + 2.11·6-s − 2.99·7-s − 1.00·8-s + 9-s − 4.67·10-s − 4.68·11-s − 2.47·12-s − 6.95·13-s + 6.33·14-s − 2.21·15-s − 2.82·16-s + 17-s − 2.11·18-s − 3.64·19-s + 5.47·20-s + 2.99·21-s + 9.91·22-s + 2.77·23-s + 1.00·24-s − 0.108·25-s + 14.7·26-s − 27-s − 7.40·28-s + ⋯
L(s)  = 1  − 1.49·2-s − 0.577·3-s + 1.23·4-s + 0.989·5-s + 0.863·6-s − 1.13·7-s − 0.354·8-s + 0.333·9-s − 1.47·10-s − 1.41·11-s − 0.714·12-s − 1.92·13-s + 1.69·14-s − 0.571·15-s − 0.706·16-s + 0.242·17-s − 0.498·18-s − 0.836·19-s + 1.22·20-s + 0.653·21-s + 2.11·22-s + 0.578·23-s + 0.204·24-s − 0.0216·25-s + 2.88·26-s − 0.192·27-s − 1.39·28-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  =  0
Selberg data  =  $(2,\ 8007,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.02670660082$
$L(\frac12)$  $\approx$  $0.02670660082$
$L(\frac{3}{2})$   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 \( 1 + 2.11T + 2T^{2} \)
5 \( 1 - 2.21T + 5T^{2} \)
7 \( 1 + 2.99T + 7T^{2} \)
11 \( 1 + 4.68T + 11T^{2} \)
13 \( 1 + 6.95T + 13T^{2} \)
19 \( 1 + 3.64T + 19T^{2} \)
23 \( 1 - 2.77T + 23T^{2} \)
29 \( 1 + 3.63T + 29T^{2} \)
31 \( 1 - 4.06T + 31T^{2} \)
37 \( 1 + 8.99T + 37T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 + 5.09T + 43T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 - 2.12T + 53T^{2} \)
59 \( 1 + 12.0T + 59T^{2} \)
61 \( 1 - 2.48T + 61T^{2} \)
67 \( 1 + 5.59T + 67T^{2} \)
71 \( 1 + 4.99T + 71T^{2} \)
73 \( 1 + 16.7T + 73T^{2} \)
79 \( 1 + 7.56T + 79T^{2} \)
83 \( 1 + 7.89T + 83T^{2} \)
89 \( 1 + 3.95T + 89T^{2} \)
97 \( 1 - 6.85T + 97T^{2} \)
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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

−7.76238766509780856596146875160, −7.20031680686801903172583734358, −6.73112598483285923435217876692, −5.87560719455445510264902361260, −5.23520723932228118595156184714, −4.53642993676464255912882696479, −3.04411530508229820476291124282, −2.39798558819756585895714364738, −1.63244898534253707873274319498, −0.099977396636088928036136082541, 0.099977396636088928036136082541, 1.63244898534253707873274319498, 2.39798558819756585895714364738, 3.04411530508229820476291124282, 4.53642993676464255912882696479, 5.23520723932228118595156184714, 5.87560719455445510264902361260, 6.73112598483285923435217876692, 7.20031680686801903172583734358, 7.76238766509780856596146875160

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