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  + 0.569·2-s + 3-s − 1.67·4-s + 1.10·5-s + 0.569·6-s + 2.67·7-s − 2.09·8-s + 9-s + 0.627·10-s − 5.58·11-s − 1.67·12-s + 5.66·13-s + 1.52·14-s + 1.10·15-s + 2.16·16-s − 17-s + 0.569·18-s − 8.17·19-s − 1.84·20-s + 2.67·21-s − 3.17·22-s + 1.82·23-s − 2.09·24-s − 3.78·25-s + 3.22·26-s + 27-s − 4.47·28-s + ⋯
L(s)  = 1  + 0.402·2-s + 0.577·3-s − 0.837·4-s + 0.492·5-s + 0.232·6-s + 1.00·7-s − 0.739·8-s + 0.333·9-s + 0.198·10-s − 1.68·11-s − 0.483·12-s + 1.57·13-s + 0.406·14-s + 0.284·15-s + 0.540·16-s − 0.242·17-s + 0.134·18-s − 1.87·19-s − 0.412·20-s + 0.582·21-s − 0.677·22-s + 0.380·23-s − 0.427·24-s − 0.757·25-s + 0.632·26-s + 0.192·27-s − 0.845·28-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 \( 1 - 0.569T + 2T^{2} \)
5 \( 1 - 1.10T + 5T^{2} \)
7 \( 1 - 2.67T + 7T^{2} \)
11 \( 1 + 5.58T + 11T^{2} \)
13 \( 1 - 5.66T + 13T^{2} \)
19 \( 1 + 8.17T + 19T^{2} \)
23 \( 1 - 1.82T + 23T^{2} \)
29 \( 1 + 0.370T + 29T^{2} \)
31 \( 1 + 0.983T + 31T^{2} \)
37 \( 1 + 7.61T + 37T^{2} \)
41 \( 1 + 0.900T + 41T^{2} \)
43 \( 1 + 0.382T + 43T^{2} \)
47 \( 1 - 11.8T + 47T^{2} \)
53 \( 1 + 6.33T + 53T^{2} \)
59 \( 1 - 2.76T + 59T^{2} \)
61 \( 1 - 4.73T + 61T^{2} \)
67 \( 1 + 15.6T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 + 3.19T + 73T^{2} \)
79 \( 1 - 9.43T + 79T^{2} \)
83 \( 1 + 6.69T + 83T^{2} \)
89 \( 1 + 5.70T + 89T^{2} \)
97 \( 1 + 6.95T + 97T^{2} \)
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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

−7.79411921888586295371154847725, −6.73002342348887571545139910548, −5.81062443363066957166057680410, −5.43296051914952334856288787494, −4.53738376140752472925340186535, −4.07693095236051476199066589441, −3.15080084715088678667499973148, −2.27413020728568630688304868695, −1.47005380843801800770496788214, 0, 1.47005380843801800770496788214, 2.27413020728568630688304868695, 3.15080084715088678667499973148, 4.07693095236051476199066589441, 4.53738376140752472925340186535, 5.43296051914952334856288787494, 5.81062443363066957166057680410, 6.73002342348887571545139910548, 7.79411921888586295371154847725

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