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  − 1.44·2-s − 3-s + 0.0754·4-s − 3.22·5-s + 1.44·6-s + 1.08·7-s + 2.77·8-s + 9-s + 4.64·10-s + 4.97·11-s − 0.0754·12-s + 1.59·13-s − 1.56·14-s + 3.22·15-s − 4.14·16-s − 17-s − 1.44·18-s + 5.94·19-s − 0.243·20-s − 1.08·21-s − 7.16·22-s + 2.41·23-s − 2.77·24-s + 5.39·25-s − 2.30·26-s − 27-s + 0.0817·28-s + ⋯
L(s)  = 1  − 1.01·2-s − 0.577·3-s + 0.0377·4-s − 1.44·5-s + 0.588·6-s + 0.409·7-s + 0.980·8-s + 0.333·9-s + 1.46·10-s + 1.49·11-s − 0.0217·12-s + 0.443·13-s − 0.417·14-s + 0.832·15-s − 1.03·16-s − 0.242·17-s − 0.339·18-s + 1.36·19-s − 0.0544·20-s − 0.236·21-s − 1.52·22-s + 0.503·23-s − 0.565·24-s + 1.07·25-s − 0.451·26-s − 0.192·27-s + 0.0154·28-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 \( 1 + 1.44T + 2T^{2} \)
5 \( 1 + 3.22T + 5T^{2} \)
7 \( 1 - 1.08T + 7T^{2} \)
11 \( 1 - 4.97T + 11T^{2} \)
13 \( 1 - 1.59T + 13T^{2} \)
19 \( 1 - 5.94T + 19T^{2} \)
23 \( 1 - 2.41T + 23T^{2} \)
29 \( 1 - 1.75T + 29T^{2} \)
31 \( 1 + 2.72T + 31T^{2} \)
37 \( 1 + 6.48T + 37T^{2} \)
41 \( 1 + 1.55T + 41T^{2} \)
43 \( 1 + 9.84T + 43T^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 + 2.06T + 53T^{2} \)
59 \( 1 - 7.21T + 59T^{2} \)
61 \( 1 - 8.92T + 61T^{2} \)
67 \( 1 + 12.8T + 67T^{2} \)
71 \( 1 - 9.11T + 71T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 - 11.7T + 79T^{2} \)
83 \( 1 - 10.6T + 83T^{2} \)
89 \( 1 - 6.95T + 89T^{2} \)
97 \( 1 + 2.86T + 97T^{2} \)
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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

−7.57842543347212801421625113340, −6.97156676793215251849423174765, −6.46912550565804945187567289141, −5.16647395013793176570677190626, −4.73666625139824406557492997969, −3.81954422965892737814081183058, −3.41178744925653285560824863582, −1.64604402684296547704042056082, −1.03169496414979039561454525913, 0, 1.03169496414979039561454525913, 1.64604402684296547704042056082, 3.41178744925653285560824863582, 3.81954422965892737814081183058, 4.73666625139824406557492997969, 5.16647395013793176570677190626, 6.46912550565804945187567289141, 6.97156676793215251849423174765, 7.57842543347212801421625113340

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