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  − 2.12·2-s + 3-s + 2.53·4-s − 4.00·5-s − 2.12·6-s + 2.85·7-s − 1.13·8-s + 9-s + 8.53·10-s + 0.813·11-s + 2.53·12-s − 0.802·13-s − 6.07·14-s − 4.00·15-s − 2.65·16-s − 17-s − 2.12·18-s + 1.48·19-s − 10.1·20-s + 2.85·21-s − 1.73·22-s + 0.00429·23-s − 1.13·24-s + 11.0·25-s + 1.70·26-s + 27-s + 7.22·28-s + ⋯
L(s)  = 1  − 1.50·2-s + 0.577·3-s + 1.26·4-s − 1.79·5-s − 0.869·6-s + 1.07·7-s − 0.400·8-s + 0.333·9-s + 2.69·10-s + 0.245·11-s + 0.731·12-s − 0.222·13-s − 1.62·14-s − 1.03·15-s − 0.663·16-s − 0.242·17-s − 0.501·18-s + 0.340·19-s − 2.26·20-s + 0.622·21-s − 0.369·22-s + 0.000894·23-s − 0.231·24-s + 2.21·25-s + 0.335·26-s + 0.192·27-s + 1.36·28-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 \( 1 + 2.12T + 2T^{2} \)
5 \( 1 + 4.00T + 5T^{2} \)
7 \( 1 - 2.85T + 7T^{2} \)
11 \( 1 - 0.813T + 11T^{2} \)
13 \( 1 + 0.802T + 13T^{2} \)
19 \( 1 - 1.48T + 19T^{2} \)
23 \( 1 - 0.00429T + 23T^{2} \)
29 \( 1 + 4.50T + 29T^{2} \)
31 \( 1 - 1.77T + 31T^{2} \)
37 \( 1 - 5.09T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + 2.58T + 43T^{2} \)
47 \( 1 + 3.09T + 47T^{2} \)
53 \( 1 - 0.598T + 53T^{2} \)
59 \( 1 - 4.12T + 59T^{2} \)
61 \( 1 - 5.03T + 61T^{2} \)
67 \( 1 + 1.96T + 67T^{2} \)
71 \( 1 - 8.01T + 71T^{2} \)
73 \( 1 - 3.22T + 73T^{2} \)
79 \( 1 - 1.07T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 - 5.84T + 89T^{2} \)
97 \( 1 + 17.9T + 97T^{2} \)
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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

−7.84106887957009022427762579801, −7.13558530336010438096397428595, −6.74241112711348495874911351388, −5.23629216616240671215172074552, −4.49009946428348511632338501583, −3.88306130373375743911508184559, −2.96747687154175068558333133042, −1.92095794799617365747409828595, −1.06358501619639505313832618390, 0, 1.06358501619639505313832618390, 1.92095794799617365747409828595, 2.96747687154175068558333133042, 3.88306130373375743911508184559, 4.49009946428348511632338501583, 5.23629216616240671215172074552, 6.74241112711348495874911351388, 7.13558530336010438096397428595, 7.84106887957009022427762579801

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