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  + 2.55·2-s − 3-s + 4.51·4-s − 2.02·5-s − 2.55·6-s − 3.46·7-s + 6.42·8-s + 9-s − 5.17·10-s + 2.24·11-s − 4.51·12-s + 2.20·13-s − 8.85·14-s + 2.02·15-s + 7.36·16-s − 17-s + 2.55·18-s + 1.23·19-s − 9.15·20-s + 3.46·21-s + 5.72·22-s − 6.14·23-s − 6.42·24-s − 0.894·25-s + 5.63·26-s − 27-s − 15.6·28-s + ⋯
L(s)  = 1  + 1.80·2-s − 0.577·3-s + 2.25·4-s − 0.906·5-s − 1.04·6-s − 1.31·7-s + 2.27·8-s + 0.333·9-s − 1.63·10-s + 0.675·11-s − 1.30·12-s + 0.612·13-s − 2.36·14-s + 0.523·15-s + 1.84·16-s − 0.242·17-s + 0.601·18-s + 0.283·19-s − 2.04·20-s + 0.757·21-s + 1.22·22-s − 1.28·23-s − 1.31·24-s − 0.178·25-s + 1.10·26-s − 0.192·27-s − 2.96·28-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 \( 1 - 2.55T + 2T^{2} \)
5 \( 1 + 2.02T + 5T^{2} \)
7 \( 1 + 3.46T + 7T^{2} \)
11 \( 1 - 2.24T + 11T^{2} \)
13 \( 1 - 2.20T + 13T^{2} \)
19 \( 1 - 1.23T + 19T^{2} \)
23 \( 1 + 6.14T + 23T^{2} \)
29 \( 1 - 6.11T + 29T^{2} \)
31 \( 1 - 7.08T + 31T^{2} \)
37 \( 1 + 8.56T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 - 0.242T + 43T^{2} \)
47 \( 1 - 5.56T + 47T^{2} \)
53 \( 1 + 2.50T + 53T^{2} \)
59 \( 1 - 2.25T + 59T^{2} \)
61 \( 1 + 5.32T + 61T^{2} \)
67 \( 1 + 4.75T + 67T^{2} \)
71 \( 1 + 7.79T + 71T^{2} \)
73 \( 1 - 9.40T + 73T^{2} \)
79 \( 1 + 2.68T + 79T^{2} \)
83 \( 1 + 7.70T + 83T^{2} \)
89 \( 1 - 1.43T + 89T^{2} \)
97 \( 1 + 8.34T + 97T^{2} \)
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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

−6.93602169953862187711323298443, −6.60698321321913221789370228354, −6.09235794301796072008152876714, −5.38493802309528658082363600305, −4.49527624537505560475341887218, −3.94807672876137643782069113264, −3.45078436816593227449014971160, −2.71793666862493182303251751469, −1.47412444659380026612186357065, 0, 1.47412444659380026612186357065, 2.71793666862493182303251751469, 3.45078436816593227449014971160, 3.94807672876137643782069113264, 4.49527624537505560475341887218, 5.38493802309528658082363600305, 6.09235794301796072008152876714, 6.60698321321913221789370228354, 6.93602169953862187711323298443

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