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  + 1.20·2-s + 3-s − 0.557·4-s − 4.37·5-s + 1.20·6-s − 0.896·7-s − 3.07·8-s + 9-s − 5.25·10-s + 0.0156·11-s − 0.557·12-s + 4.65·13-s − 1.07·14-s − 4.37·15-s − 2.57·16-s − 17-s + 1.20·18-s − 0.462·19-s + 2.43·20-s − 0.896·21-s + 0.0187·22-s + 3.14·23-s − 3.07·24-s + 14.1·25-s + 5.59·26-s + 27-s + 0.499·28-s + ⋯
L(s)  = 1  + 0.849·2-s + 0.577·3-s − 0.278·4-s − 1.95·5-s + 0.490·6-s − 0.338·7-s − 1.08·8-s + 0.333·9-s − 1.66·10-s + 0.00471·11-s − 0.160·12-s + 1.29·13-s − 0.287·14-s − 1.12·15-s − 0.643·16-s − 0.242·17-s + 0.283·18-s − 0.106·19-s + 0.544·20-s − 0.195·21-s + 0.00400·22-s + 0.656·23-s − 0.626·24-s + 2.82·25-s + 1.09·26-s + 0.192·27-s + 0.0943·28-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 \( 1 - 1.20T + 2T^{2} \)
5 \( 1 + 4.37T + 5T^{2} \)
7 \( 1 + 0.896T + 7T^{2} \)
11 \( 1 - 0.0156T + 11T^{2} \)
13 \( 1 - 4.65T + 13T^{2} \)
19 \( 1 + 0.462T + 19T^{2} \)
23 \( 1 - 3.14T + 23T^{2} \)
29 \( 1 - 6.55T + 29T^{2} \)
31 \( 1 - 1.25T + 31T^{2} \)
37 \( 1 + 2.56T + 37T^{2} \)
41 \( 1 + 5.93T + 41T^{2} \)
43 \( 1 - 4.83T + 43T^{2} \)
47 \( 1 + 6.79T + 47T^{2} \)
53 \( 1 - 2.50T + 53T^{2} \)
59 \( 1 + 6.82T + 59T^{2} \)
61 \( 1 + 14.4T + 61T^{2} \)
67 \( 1 - 4.96T + 67T^{2} \)
71 \( 1 + 8.90T + 71T^{2} \)
73 \( 1 - 3.61T + 73T^{2} \)
79 \( 1 + 7.29T + 79T^{2} \)
83 \( 1 - 12.6T + 83T^{2} \)
89 \( 1 - 11.7T + 89T^{2} \)
97 \( 1 - 9.59T + 97T^{2} \)
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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

−7.63170443242083398776458253917, −6.65655093514046675845949120412, −6.26550208922882941529307682286, −4.95931802294642369512775125330, −4.58621934886310537664092983120, −3.76923830844477246670934490310, −3.40562253207430301330463610214, −2.81124601592579582424736361702, −1.12832845045148973975759589311, 0, 1.12832845045148973975759589311, 2.81124601592579582424736361702, 3.40562253207430301330463610214, 3.76923830844477246670934490310, 4.58621934886310537664092983120, 4.95931802294642369512775125330, 6.26550208922882941529307682286, 6.65655093514046675845949120412, 7.63170443242083398776458253917

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