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  + 0.522·2-s − 3-s − 1.72·4-s − 2.49·5-s − 0.522·6-s + 5.20·7-s − 1.94·8-s + 9-s − 1.30·10-s + 3.79·11-s + 1.72·12-s + 1.98·13-s + 2.71·14-s + 2.49·15-s + 2.43·16-s − 17-s + 0.522·18-s − 4.44·19-s + 4.30·20-s − 5.20·21-s + 1.98·22-s + 1.01·23-s + 1.94·24-s + 1.20·25-s + 1.03·26-s − 27-s − 8.99·28-s + ⋯
L(s)  = 1  + 0.369·2-s − 0.577·3-s − 0.863·4-s − 1.11·5-s − 0.213·6-s + 1.96·7-s − 0.688·8-s + 0.333·9-s − 0.411·10-s + 1.14·11-s + 0.498·12-s + 0.551·13-s + 0.726·14-s + 0.643·15-s + 0.609·16-s − 0.242·17-s + 0.123·18-s − 1.02·19-s + 0.962·20-s − 1.13·21-s + 0.422·22-s + 0.211·23-s + 0.397·24-s + 0.241·25-s + 0.203·26-s − 0.192·27-s − 1.69·28-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 \( 1 - 0.522T + 2T^{2} \)
5 \( 1 + 2.49T + 5T^{2} \)
7 \( 1 - 5.20T + 7T^{2} \)
11 \( 1 - 3.79T + 11T^{2} \)
13 \( 1 - 1.98T + 13T^{2} \)
19 \( 1 + 4.44T + 19T^{2} \)
23 \( 1 - 1.01T + 23T^{2} \)
29 \( 1 + 8.84T + 29T^{2} \)
31 \( 1 + 7.21T + 31T^{2} \)
37 \( 1 + 11.2T + 37T^{2} \)
41 \( 1 - 3.60T + 41T^{2} \)
43 \( 1 - 3.48T + 43T^{2} \)
47 \( 1 + 0.328T + 47T^{2} \)
53 \( 1 - 7.74T + 53T^{2} \)
59 \( 1 - 9.57T + 59T^{2} \)
61 \( 1 + 4.50T + 61T^{2} \)
67 \( 1 - 1.51T + 67T^{2} \)
71 \( 1 - 0.798T + 71T^{2} \)
73 \( 1 + 2.42T + 73T^{2} \)
79 \( 1 + 0.0708T + 79T^{2} \)
83 \( 1 + 6.45T + 83T^{2} \)
89 \( 1 + 9.71T + 89T^{2} \)
97 \( 1 - 6.02T + 97T^{2} \)
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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

−7.51965737925393891314696428140, −6.89875223588439371147711237949, −5.79874598571287841815394315434, −5.32103324108462708217385582992, −4.55462224933259444498601036446, −3.93436031686338868749338808295, −3.75684637801509698336824522378, −2.00443860192192522690979072383, −1.17884944952003147370012520732, 0, 1.17884944952003147370012520732, 2.00443860192192522690979072383, 3.75684637801509698336824522378, 3.93436031686338868749338808295, 4.55462224933259444498601036446, 5.32103324108462708217385582992, 5.79874598571287841815394315434, 6.89875223588439371147711237949, 7.51965737925393891314696428140

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