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  − 0.240·2-s − 3-s − 1.94·4-s − 0.991·5-s + 0.240·6-s + 2.53·7-s + 0.946·8-s + 9-s + 0.237·10-s + 0.968·11-s + 1.94·12-s + 2.48·13-s − 0.608·14-s + 0.991·15-s + 3.65·16-s − 17-s − 0.240·18-s − 5.03·19-s + 1.92·20-s − 2.53·21-s − 0.232·22-s − 1.80·23-s − 0.946·24-s − 4.01·25-s − 0.597·26-s − 27-s − 4.92·28-s + ⋯
L(s)  = 1  − 0.169·2-s − 0.577·3-s − 0.971·4-s − 0.443·5-s + 0.0979·6-s + 0.958·7-s + 0.334·8-s + 0.333·9-s + 0.0752·10-s + 0.291·11-s + 0.560·12-s + 0.690·13-s − 0.162·14-s + 0.255·15-s + 0.914·16-s − 0.242·17-s − 0.0565·18-s − 1.15·19-s + 0.430·20-s − 0.553·21-s − 0.0495·22-s − 0.376·23-s − 0.193·24-s − 0.803·25-s − 0.117·26-s − 0.192·27-s − 0.930·28-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
17 \( 1 + T \)
157 \( 1 + T \)
good2 \( 1 + 0.240T + 2T^{2} \)
5 \( 1 + 0.991T + 5T^{2} \)
7 \( 1 - 2.53T + 7T^{2} \)
11 \( 1 - 0.968T + 11T^{2} \)
13 \( 1 - 2.48T + 13T^{2} \)
19 \( 1 + 5.03T + 19T^{2} \)
23 \( 1 + 1.80T + 23T^{2} \)
29 \( 1 - 5.21T + 29T^{2} \)
31 \( 1 - 6.87T + 31T^{2} \)
37 \( 1 - 2.49T + 37T^{2} \)
41 \( 1 + 1.02T + 41T^{2} \)
43 \( 1 + 9.96T + 43T^{2} \)
47 \( 1 + 3.98T + 47T^{2} \)
53 \( 1 + 5.76T + 53T^{2} \)
59 \( 1 - 0.764T + 59T^{2} \)
61 \( 1 + 6.15T + 61T^{2} \)
67 \( 1 - 0.538T + 67T^{2} \)
71 \( 1 - 9.03T + 71T^{2} \)
73 \( 1 + 2.45T + 73T^{2} \)
79 \( 1 - 7.60T + 79T^{2} \)
83 \( 1 - 2.86T + 83T^{2} \)
89 \( 1 + 0.260T + 89T^{2} \)
97 \( 1 - 11.8T + 97T^{2} \)
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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

−7.81141158391993484660569200128, −6.62367275053127990566852033022, −6.19294239510622006545615136461, −5.21724081109937241260330119094, −4.59453066322395359120902210013, −4.18570456283033091978299325801, −3.33732011406215158171730712065, −1.94877155907505196545291466954, −1.08617859617998330349342599052, 0, 1.08617859617998330349342599052, 1.94877155907505196545291466954, 3.33732011406215158171730712065, 4.18570456283033091978299325801, 4.59453066322395359120902210013, 5.21724081109937241260330119094, 6.19294239510622006545615136461, 6.62367275053127990566852033022, 7.81141158391993484660569200128

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