Properties

Degree $2$
Conductor $1008$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 7-s − 6·11-s − 6·13-s − 2·17-s − 4·19-s − 2·23-s − 25-s + 8·29-s − 4·31-s − 2·35-s − 6·37-s + 10·41-s + 4·43-s + 4·47-s + 49-s − 4·53-s − 12·55-s + 12·59-s − 2·61-s − 12·65-s − 12·67-s − 6·71-s − 2·73-s + 6·77-s + 8·79-s − 4·85-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.377·7-s − 1.80·11-s − 1.66·13-s − 0.485·17-s − 0.917·19-s − 0.417·23-s − 1/5·25-s + 1.48·29-s − 0.718·31-s − 0.338·35-s − 0.986·37-s + 1.56·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s − 0.549·53-s − 1.61·55-s + 1.56·59-s − 0.256·61-s − 1.48·65-s − 1.46·67-s − 0.712·71-s − 0.234·73-s + 0.683·77-s + 0.900·79-s − 0.433·85-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

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{1008} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.54528897277219, −19.36381501089846, −18.26534413177125, −17.69582760985190, −17.24914490691715, −16.25125414055775, −15.69452575905813, −14.87350043879913, −14.09902171360345, −13.37730832037232, −12.70541185874077, −12.18055217512072, −10.86425820731114, −10.27151701104803, −9.741330569049992, −8.799787487059595, −7.815742956443090, −7.063498625491902, −6.010345845238724, −5.265014183436182, −4.396047006454253, −2.731836633848797, −2.213218596510132, 0, 2.213218596510132, 2.731836633848797, 4.396047006454253, 5.265014183436182, 6.010345845238724, 7.063498625491902, 7.815742956443090, 8.799787487059595, 9.741330569049992, 10.27151701104803, 10.86425820731114, 12.18055217512072, 12.70541185874077, 13.37730832037232, 14.09902171360345, 14.87350043879913, 15.69452575905813, 16.25125414055775, 17.24914490691715, 17.69582760985190, 18.26534413177125, 19.36381501089846, 19.54528897277219

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