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  + 1.40·2-s − 3-s − 0.0266·4-s + 2.64·5-s − 1.40·6-s − 2.79·7-s − 2.84·8-s + 9-s + 3.71·10-s + 1.62·11-s + 0.0266·12-s + 0.165·13-s − 3.92·14-s − 2.64·15-s − 3.94·16-s − 17-s + 1.40·18-s − 0.0799·19-s − 0.0705·20-s + 2.79·21-s + 2.27·22-s + 4.53·23-s + 2.84·24-s + 2.01·25-s + 0.232·26-s − 27-s + 0.0744·28-s + ⋯
L(s)  = 1  + 0.993·2-s − 0.577·3-s − 0.0133·4-s + 1.18·5-s − 0.573·6-s − 1.05·7-s − 1.00·8-s + 0.333·9-s + 1.17·10-s + 0.488·11-s + 0.00769·12-s + 0.0459·13-s − 1.04·14-s − 0.683·15-s − 0.986·16-s − 0.242·17-s + 0.331·18-s − 0.0183·19-s − 0.0157·20-s + 0.609·21-s + 0.485·22-s + 0.944·23-s + 0.581·24-s + 0.402·25-s + 0.0456·26-s − 0.192·27-s + 0.0140·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\n\]

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 \( 1 - 1.40T + 2T^{2} \)
5 \( 1 - 2.64T + 5T^{2} \)
7 \( 1 + 2.79T + 7T^{2} \)
11 \( 1 - 1.62T + 11T^{2} \)
13 \( 1 - 0.165T + 13T^{2} \)
19 \( 1 + 0.0799T + 19T^{2} \)
23 \( 1 - 4.53T + 23T^{2} \)
29 \( 1 + 2.83T + 29T^{2} \)
31 \( 1 - 4.25T + 31T^{2} \)
37 \( 1 + 3.78T + 37T^{2} \)
41 \( 1 - 8.76T + 41T^{2} \)
43 \( 1 + 12.2T + 43T^{2} \)
47 \( 1 - 3.98T + 47T^{2} \)
53 \( 1 + 10.2T + 53T^{2} \)
59 \( 1 + 8.23T + 59T^{2} \)
61 \( 1 - 8.41T + 61T^{2} \)
67 \( 1 - 5.72T + 67T^{2} \)
71 \( 1 - 7.22T + 71T^{2} \)
73 \( 1 + 0.0542T + 73T^{2} \)
79 \( 1 + 4.31T + 79T^{2} \)
83 \( 1 + 7.16T + 83T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 - 2.43T + 97T^{2} \)
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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

−6.95311596873816556244930724380, −6.55624000186437970203127648997, −6.00903686556754879530678490067, −5.43307953038293313480389902919, −4.80883433076308310918690309883, −3.97973401553010792221293533756, −3.22198956334987079783790086686, −2.47236289140882840868368422254, −1.32342276617596377272203778189, 0, 1.32342276617596377272203778189, 2.47236289140882840868368422254, 3.22198956334987079783790086686, 3.97973401553010792221293533756, 4.80883433076308310918690309883, 5.43307953038293313480389902919, 6.00903686556754879530678490067, 6.55624000186437970203127648997, 6.95311596873816556244930724380

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