Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 7-s + 6·13-s + 2·17-s − 4·19-s − 4·23-s − 25-s + 10·29-s + 8·31-s + 2·35-s + 6·37-s + 2·41-s + 4·43-s + 8·47-s + 49-s + 10·53-s + 12·59-s − 2·61-s − 12·65-s − 12·67-s − 12·71-s − 14·73-s + 8·79-s + 12·83-s − 4·85-s + 2·89-s − 6·91-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s + 1.66·13-s + 0.485·17-s − 0.917·19-s − 0.834·23-s − 1/5·25-s + 1.85·29-s + 1.43·31-s + 0.338·35-s + 0.986·37-s + 0.312·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 1.37·53-s + 1.56·59-s − 0.256·61-s − 1.48·65-s − 1.46·67-s − 1.42·71-s − 1.63·73-s + 0.900·79-s + 1.31·83-s − 0.433·85-s + 0.211·89-s − 0.628·91-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.340324021$
$L(\frac12)$  $\approx$  $1.340324021$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−19.34858529726055, −19.00377005584225, −18.08369559362299, −17.50757365402609, −16.40046243847256, −15.98504412017359, −15.44585851597152, −14.58615197703788, −13.67804655985953, −13.15080770886273, −12.03186467838566, −11.76944116855058, −10.64252841247664, −10.18277858195260, −8.913562206930639, −8.334199259994988, −7.600263079779829, −6.437980125783011, −5.909708293452956, −4.400341478944412, −3.829315989426389, −2.674935040759643, −0.9259741509304846, 0.9259741509304846, 2.674935040759643, 3.829315989426389, 4.400341478944412, 5.909708293452956, 6.437980125783011, 7.600263079779829, 8.334199259994988, 8.913562206930639, 10.18277858195260, 10.64252841247664, 11.76944116855058, 12.03186467838566, 13.15080770886273, 13.67804655985953, 14.58615197703788, 15.44585851597152, 15.98504412017359, 16.40046243847256, 17.50757365402609, 18.08369559362299, 19.00377005584225, 19.34858529726055

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