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  − 1.50·2-s − 3-s + 0.270·4-s − 0.475·5-s + 1.50·6-s − 0.891·7-s + 2.60·8-s + 9-s + 0.716·10-s − 4.53·11-s − 0.270·12-s + 1.62·13-s + 1.34·14-s + 0.475·15-s − 4.46·16-s − 17-s − 1.50·18-s + 0.833·19-s − 0.128·20-s + 0.891·21-s + 6.82·22-s − 3.25·23-s − 2.60·24-s − 4.77·25-s − 2.44·26-s − 27-s − 0.240·28-s + ⋯
L(s)  = 1  − 1.06·2-s − 0.577·3-s + 0.135·4-s − 0.212·5-s + 0.615·6-s − 0.337·7-s + 0.921·8-s + 0.333·9-s + 0.226·10-s − 1.36·11-s − 0.0779·12-s + 0.450·13-s + 0.359·14-s + 0.122·15-s − 1.11·16-s − 0.242·17-s − 0.355·18-s + 0.191·19-s − 0.0287·20-s + 0.194·21-s + 1.45·22-s − 0.678·23-s − 0.532·24-s − 0.954·25-s − 0.479·26-s − 0.192·27-s − 0.0455·28-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 \( 1 + 1.50T + 2T^{2} \)
5 \( 1 + 0.475T + 5T^{2} \)
7 \( 1 + 0.891T + 7T^{2} \)
11 \( 1 + 4.53T + 11T^{2} \)
13 \( 1 - 1.62T + 13T^{2} \)
19 \( 1 - 0.833T + 19T^{2} \)
23 \( 1 + 3.25T + 23T^{2} \)
29 \( 1 - 4.74T + 29T^{2} \)
31 \( 1 - 10.3T + 31T^{2} \)
37 \( 1 + 4.87T + 37T^{2} \)
41 \( 1 - 4.26T + 41T^{2} \)
43 \( 1 + 9.65T + 43T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 - 0.534T + 53T^{2} \)
59 \( 1 - 12.6T + 59T^{2} \)
61 \( 1 - 0.228T + 61T^{2} \)
67 \( 1 - 8.20T + 67T^{2} \)
71 \( 1 + 8.50T + 71T^{2} \)
73 \( 1 + 6.38T + 73T^{2} \)
79 \( 1 + 4.83T + 79T^{2} \)
83 \( 1 + 17.0T + 83T^{2} \)
89 \( 1 + 1.87T + 89T^{2} \)
97 \( 1 + 8.80T + 97T^{2} \)
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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

−7.58977267395593937764115021498, −7.00691625098578892291742905580, −6.18165222716728960297295961031, −5.46856017702573793286838880496, −4.68241785441035456960143189073, −4.04393976290095595045483156236, −2.93393518917477279497367374148, −1.99030167393094210638158867651, −0.865632464362979667285869979046, 0, 0.865632464362979667285869979046, 1.99030167393094210638158867651, 2.93393518917477279497367374148, 4.04393976290095595045483156236, 4.68241785441035456960143189073, 5.46856017702573793286838880496, 6.18165222716728960297295961031, 7.00691625098578892291742905580, 7.58977267395593937764115021498

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