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  + 1.82·2-s + 3-s + 1.34·4-s − 2.81·5-s + 1.82·6-s − 0.871·7-s − 1.19·8-s + 9-s − 5.15·10-s + 3.43·11-s + 1.34·12-s − 1.85·13-s − 1.59·14-s − 2.81·15-s − 4.87·16-s − 17-s + 1.82·18-s + 6.88·19-s − 3.79·20-s − 0.871·21-s + 6.27·22-s − 1.42·23-s − 1.19·24-s + 2.94·25-s − 3.38·26-s + 27-s − 1.17·28-s + ⋯
L(s)  = 1  + 1.29·2-s + 0.577·3-s + 0.673·4-s − 1.26·5-s + 0.746·6-s − 0.329·7-s − 0.422·8-s + 0.333·9-s − 1.63·10-s + 1.03·11-s + 0.388·12-s − 0.513·13-s − 0.426·14-s − 0.727·15-s − 1.21·16-s − 0.242·17-s + 0.431·18-s + 1.57·19-s − 0.849·20-s − 0.190·21-s + 1.33·22-s − 0.296·23-s − 0.243·24-s + 0.588·25-s − 0.664·26-s + 0.192·27-s − 0.222·28-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 \( 1 - 1.82T + 2T^{2} \)
5 \( 1 + 2.81T + 5T^{2} \)
7 \( 1 + 0.871T + 7T^{2} \)
11 \( 1 - 3.43T + 11T^{2} \)
13 \( 1 + 1.85T + 13T^{2} \)
19 \( 1 - 6.88T + 19T^{2} \)
23 \( 1 + 1.42T + 23T^{2} \)
29 \( 1 + 0.416T + 29T^{2} \)
31 \( 1 - 6.13T + 31T^{2} \)
37 \( 1 - 3.31T + 37T^{2} \)
41 \( 1 + 1.31T + 41T^{2} \)
43 \( 1 + 5.48T + 43T^{2} \)
47 \( 1 + 7.78T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + 8.69T + 59T^{2} \)
61 \( 1 - 7.59T + 61T^{2} \)
67 \( 1 - 4.54T + 67T^{2} \)
71 \( 1 - 0.460T + 71T^{2} \)
73 \( 1 + 9.18T + 73T^{2} \)
79 \( 1 + 6.27T + 79T^{2} \)
83 \( 1 + 9.38T + 83T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 + 15.4T + 97T^{2} \)
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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

−7.35818626541491765543983495343, −6.74185976764516247787598352102, −6.10900212387594677255039577419, −5.09766476272367704780059087240, −4.50646775802988646955404481550, −3.91603911053664672738687268375, −3.26236956268951607280378139610, −2.81924220716418754054921757974, −1.44002427701087206051243011757, 0, 1.44002427701087206051243011757, 2.81924220716418754054921757974, 3.26236956268951607280378139610, 3.91603911053664672738687268375, 4.50646775802988646955404481550, 5.09766476272367704780059087240, 6.10900212387594677255039577419, 6.74185976764516247787598352102, 7.35818626541491765543983495343

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