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  − 1.41·2-s + 3-s + 0.00726·4-s + 2.31·5-s − 1.41·6-s + 2.67·7-s + 2.82·8-s + 9-s − 3.27·10-s − 4.03·11-s + 0.00726·12-s − 3.81·13-s − 3.79·14-s + 2.31·15-s − 4.01·16-s − 17-s − 1.41·18-s − 5.43·19-s + 0.0167·20-s + 2.67·21-s + 5.71·22-s − 4.96·23-s + 2.82·24-s + 0.339·25-s + 5.40·26-s + 27-s + 0.0194·28-s + ⋯
L(s)  = 1  − 1.00·2-s + 0.577·3-s + 0.00363·4-s + 1.03·5-s − 0.578·6-s + 1.01·7-s + 0.998·8-s + 0.333·9-s − 1.03·10-s − 1.21·11-s + 0.00209·12-s − 1.05·13-s − 1.01·14-s + 0.596·15-s − 1.00·16-s − 0.242·17-s − 0.333·18-s − 1.24·19-s + 0.00375·20-s + 0.584·21-s + 1.21·22-s − 1.03·23-s + 0.576·24-s + 0.0678·25-s + 1.06·26-s + 0.192·27-s + 0.00367·28-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 \( 1 + 1.41T + 2T^{2} \)
5 \( 1 - 2.31T + 5T^{2} \)
7 \( 1 - 2.67T + 7T^{2} \)
11 \( 1 + 4.03T + 11T^{2} \)
13 \( 1 + 3.81T + 13T^{2} \)
19 \( 1 + 5.43T + 19T^{2} \)
23 \( 1 + 4.96T + 23T^{2} \)
29 \( 1 - 7.33T + 29T^{2} \)
31 \( 1 - 7.79T + 31T^{2} \)
37 \( 1 + 4.21T + 37T^{2} \)
41 \( 1 + 0.767T + 41T^{2} \)
43 \( 1 + 0.0132T + 43T^{2} \)
47 \( 1 + 4.65T + 47T^{2} \)
53 \( 1 - 7.15T + 53T^{2} \)
59 \( 1 + 1.07T + 59T^{2} \)
61 \( 1 - 5.20T + 61T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 - 8.38T + 71T^{2} \)
73 \( 1 + 15.9T + 73T^{2} \)
79 \( 1 - 13.7T + 79T^{2} \)
83 \( 1 + 5.99T + 83T^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−7.889403477154054819798872955416, −7.01400209575136821539316050876, −6.27554688004160904226124112234, −5.16351780092602078947359077621, −4.83714546598012313014539627980, −4.02635990947359875259275830017, −2.45287375665350978622690069550, −2.28691419607285577966691080768, −1.32787373782058694017965704521, 0, 1.32787373782058694017965704521, 2.28691419607285577966691080768, 2.45287375665350978622690069550, 4.02635990947359875259275830017, 4.83714546598012313014539627980, 5.16351780092602078947359077621, 6.27554688004160904226124112234, 7.01400209575136821539316050876, 7.889403477154054819798872955416

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