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: $\chi_{1008} (1, \cdot )$
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

−9.411301062945376474304956125779, −8.895731435417720392977847601424, −7.71183835160919220569405836746, −7.11617364040573412059356557368, −6.28688515386336055288485317654, −5.07010871027621008671481451395, −4.07111090082168476326491648685, −3.36279226696466149410004076881, −1.83565519602500234078552844880, 0, 1.83565519602500234078552844880, 3.36279226696466149410004076881, 4.07111090082168476326491648685, 5.07010871027621008671481451395, 6.28688515386336055288485317654, 7.11617364040573412059356557368, 7.71183835160919220569405836746, 8.895731435417720392977847601424, 9.411301062945376474304956125779

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