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  + 1.36·2-s − 3-s − 0.149·4-s − 3.22·5-s − 1.36·6-s + 2.99·7-s − 2.92·8-s + 9-s − 4.39·10-s − 3.10·11-s + 0.149·12-s + 1.97·13-s + 4.07·14-s + 3.22·15-s − 3.67·16-s − 17-s + 1.36·18-s − 0.710·19-s + 0.483·20-s − 2.99·21-s − 4.21·22-s + 6.91·23-s + 2.92·24-s + 5.42·25-s + 2.69·26-s − 27-s − 0.449·28-s + ⋯
L(s)  = 1  + 0.961·2-s − 0.577·3-s − 0.0749·4-s − 1.44·5-s − 0.555·6-s + 1.13·7-s − 1.03·8-s + 0.333·9-s − 1.38·10-s − 0.934·11-s + 0.0432·12-s + 0.549·13-s + 1.08·14-s + 0.833·15-s − 0.919·16-s − 0.242·17-s + 0.320·18-s − 0.163·19-s + 0.108·20-s − 0.653·21-s − 0.899·22-s + 1.44·23-s + 0.596·24-s + 1.08·25-s + 0.528·26-s − 0.192·27-s − 0.0848·28-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 \( 1 - 1.36T + 2T^{2} \)
5 \( 1 + 3.22T + 5T^{2} \)
7 \( 1 - 2.99T + 7T^{2} \)
11 \( 1 + 3.10T + 11T^{2} \)
13 \( 1 - 1.97T + 13T^{2} \)
19 \( 1 + 0.710T + 19T^{2} \)
23 \( 1 - 6.91T + 23T^{2} \)
29 \( 1 - 9.35T + 29T^{2} \)
31 \( 1 + 7.97T + 31T^{2} \)
37 \( 1 + 4.06T + 37T^{2} \)
41 \( 1 + 0.959T + 41T^{2} \)
43 \( 1 + 1.47T + 43T^{2} \)
47 \( 1 - 5.23T + 47T^{2} \)
53 \( 1 - 4.09T + 53T^{2} \)
59 \( 1 - 2.47T + 59T^{2} \)
61 \( 1 - 8.55T + 61T^{2} \)
67 \( 1 + 5.39T + 67T^{2} \)
71 \( 1 + 0.392T + 71T^{2} \)
73 \( 1 + 3.98T + 73T^{2} \)
79 \( 1 + 5.00T + 79T^{2} \)
83 \( 1 - 13.6T + 83T^{2} \)
89 \( 1 - 9.10T + 89T^{2} \)
97 \( 1 + 15.0T + 97T^{2} \)
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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

−7.33932062400207623165141215093, −6.84158216366765407659934038811, −5.82670491604275361720728763269, −5.06907849221385189041977630410, −4.80429562004053498327370913379, −4.05533224488613150701577736425, −3.42826714891881909018240378314, −2.51030955344667695255439805346, −1.08059591528998224626007079922, 0, 1.08059591528998224626007079922, 2.51030955344667695255439805346, 3.42826714891881909018240378314, 4.05533224488613150701577736425, 4.80429562004053498327370913379, 5.06907849221385189041977630410, 5.82670491604275361720728763269, 6.84158216366765407659934038811, 7.33932062400207623165141215093

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