Properties

Degree $2$
Conductor $4002$
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-s − 3-s + 4-s + 5-s − 6-s + 7-s + 8-s + 9-s + 10-s − 4·11-s − 12-s − 2·13-s + 14-s − 15-s + 16-s − 3·17-s + 18-s + 19-s + 20-s − 21-s − 4·22-s + 23-s − 24-s − 4·25-s − 2·26-s − 27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.218·21-s − 0.852·22-s + 0.208·23-s − 0.204·24-s − 4/5·25-s − 0.392·26-s − 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4002\)    =    \(2 \cdot 3 \cdot 23 \cdot 29\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{4002} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4002,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 + T \)
good5 \( 1 - T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−18.27214563279074, −17.84227752946958, −17.08607439443567, −16.66700680946505, −15.79743718608936, −15.35149986836017, −14.82776912659400, −13.83561379078386, −13.52057941104322, −12.86377664147116, −12.19075528920225, −11.59430099934337, −10.88080463545722, −10.36084729558461, −9.700893890527362, −8.776004288299583, −7.833217681924268, −7.287171887939288, −6.441666209209084, −5.707603917753274, −5.071067924031934, −4.573521532298667, −3.458073981331447, −2.458139308963358, −1.668174515771830, 0, 1.668174515771830, 2.458139308963358, 3.458073981331447, 4.573521532298667, 5.071067924031934, 5.707603917753274, 6.441666209209084, 7.287171887939288, 7.833217681924268, 8.776004288299583, 9.700893890527362, 10.36084729558461, 10.88080463545722, 11.59430099934337, 12.19075528920225, 12.86377664147116, 13.52057941104322, 13.83561379078386, 14.82776912659400, 15.35149986836017, 15.79743718608936, 16.66700680946505, 17.08607439443567, 17.84227752946958, 18.27214563279074

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