Properties

Degree $2$
Conductor $52983$
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 − 4-s + 2·5-s + 7-s + 3·8-s − 2·10-s + 4·11-s − 2·13-s − 14-s − 16-s + 2·17-s + 4·19-s − 2·20-s − 4·22-s − 25-s + 2·26-s − 28-s + 8·31-s − 5·32-s − 2·34-s + 2·35-s + 10·37-s − 4·38-s + 6·40-s − 6·41-s − 12·43-s − 4·44-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s + 1.06·8-s − 0.632·10-s + 1.20·11-s − 0.554·13-s − 0.267·14-s − 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.447·20-s − 0.852·22-s − 1/5·25-s + 0.392·26-s − 0.188·28-s + 1.43·31-s − 0.883·32-s − 0.342·34-s + 0.338·35-s + 1.64·37-s − 0.648·38-s + 0.948·40-s − 0.937·41-s − 1.82·43-s − 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52983\)    =    \(3^{2} \cdot 7 \cdot 29^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{52983} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 52983,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
7 \( 1 - T \)
29 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.61747958246541, −14.05315717194155, −13.86301972236002, −13.26148952307112, −12.79094750542312, −11.95365484802472, −11.68709176811857, −11.11471497672079, −10.27292843047557, −9.850568650221929, −9.661409617030266, −9.137432670514625, −8.497271061518263, −7.964219356040626, −7.579161429217394, −6.694380487468667, −6.366342141279308, −5.588225651440327, −4.947527876228492, −4.607966962584900, −3.788914150172916, −3.127683358205314, −2.249594114384621, −1.387920190348360, −1.189162938331079, 0, 1.189162938331079, 1.387920190348360, 2.249594114384621, 3.127683358205314, 3.788914150172916, 4.607966962584900, 4.947527876228492, 5.588225651440327, 6.366342141279308, 6.694380487468667, 7.579161429217394, 7.964219356040626, 8.497271061518263, 9.137432670514625, 9.661409617030266, 9.850568650221929, 10.27292843047557, 11.11471497672079, 11.68709176811857, 11.95365484802472, 12.79094750542312, 13.26148952307112, 13.86301972236002, 14.05315717194155, 14.61747958246541

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