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  + 0.941·2-s − 3-s − 1.11·4-s − 2.91·5-s − 0.941·6-s − 1.76·7-s − 2.93·8-s + 9-s − 2.74·10-s − 4.88·11-s + 1.11·12-s − 2.39·13-s − 1.65·14-s + 2.91·15-s − 0.531·16-s − 17-s + 0.941·18-s + 6.33·19-s + 3.24·20-s + 1.76·21-s − 4.60·22-s + 3.09·23-s + 2.93·24-s + 3.48·25-s − 2.25·26-s − 27-s + 1.96·28-s + ⋯
L(s)  = 1  + 0.665·2-s − 0.577·3-s − 0.556·4-s − 1.30·5-s − 0.384·6-s − 0.665·7-s − 1.03·8-s + 0.333·9-s − 0.867·10-s − 1.47·11-s + 0.321·12-s − 0.665·13-s − 0.443·14-s + 0.752·15-s − 0.132·16-s − 0.242·17-s + 0.221·18-s + 1.45·19-s + 0.725·20-s + 0.384·21-s − 0.981·22-s + 0.644·23-s + 0.598·24-s + 0.697·25-s − 0.442·26-s − 0.192·27-s + 0.370·28-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 \( 1 - 0.941T + 2T^{2} \)
5 \( 1 + 2.91T + 5T^{2} \)
7 \( 1 + 1.76T + 7T^{2} \)
11 \( 1 + 4.88T + 11T^{2} \)
13 \( 1 + 2.39T + 13T^{2} \)
19 \( 1 - 6.33T + 19T^{2} \)
23 \( 1 - 3.09T + 23T^{2} \)
29 \( 1 - 5.08T + 29T^{2} \)
31 \( 1 - 4.97T + 31T^{2} \)
37 \( 1 - 5.58T + 37T^{2} \)
41 \( 1 - 3.89T + 41T^{2} \)
43 \( 1 - 1.41T + 43T^{2} \)
47 \( 1 + 3.41T + 47T^{2} \)
53 \( 1 + 5.53T + 53T^{2} \)
59 \( 1 - 8.82T + 59T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 + 7.53T + 67T^{2} \)
71 \( 1 - 9.14T + 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 - 1.67T + 79T^{2} \)
83 \( 1 - 3.13T + 83T^{2} \)
89 \( 1 - 3.90T + 89T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−7.68205663666879387447344073483, −6.70201542027491638317350725521, −5.98094702338893072272052967143, −5.10190595576502318410598493736, −4.79820647449650463498017125493, −4.06068851472502943242515471374, −3.11159848442871096162019428054, −2.78239199288998033983606230110, −0.801706775614260952251323049293, 0, 0.801706775614260952251323049293, 2.78239199288998033983606230110, 3.11159848442871096162019428054, 4.06068851472502943242515471374, 4.79820647449650463498017125493, 5.10190595576502318410598493736, 5.98094702338893072272052967143, 6.70201542027491638317350725521, 7.68205663666879387447344073483

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