Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 23 \cdot 29 $
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-s − 3-s + 4-s + 5-s + 6-s + 4·7-s − 8-s + 9-s − 10-s − 3·11-s − 12-s + 13-s − 4·14-s − 15-s + 16-s + 6·17-s − 18-s + 6·19-s + 20-s − 4·21-s + 3·22-s − 23-s + 24-s − 4·25-s − 26-s − 27-s + 4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.904·11-s − 0.288·12-s + 0.277·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 1.37·19-s + 0.223·20-s − 0.872·21-s + 0.639·22-s − 0.208·23-s + 0.204·24-s − 4/5·25-s − 0.196·26-s − 0.192·27-s + 0.755·28-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

\( d \)  =  \(2\)
\( N \)  =  \(4002\)    =    \(2 \cdot 3 \cdot 23 \cdot 29\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4002} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4002,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.587112286$
$L(\frac12)$  $\approx$  $1.587112286$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;23,\;29\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 - T \)
good5 \( 1 - T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−18.03514871090390, −17.60196514703167, −17.11091235687797, −16.33127799441379, −15.84397997450298, −15.19048615198884, −14.27021942482859, −14.01654260985262, −13.07269285130994, −12.20159247266940, −11.75443578392543, −11.03256231174068, −10.60355036088756, −9.796214284338600, −9.344719265747599, −8.116518031188910, −7.897389736606163, −7.269015156128929, −6.126749281073745, −5.425396632897746, −5.062314170007118, −3.857754945230739, −2.685091584003517, −1.669162749962358, −0.8947855604348575, 0.8947855604348575, 1.669162749962358, 2.685091584003517, 3.857754945230739, 5.062314170007118, 5.425396632897746, 6.126749281073745, 7.269015156128929, 7.897389736606163, 8.116518031188910, 9.344719265747599, 9.796214284338600, 10.60355036088756, 11.03256231174068, 11.75443578392543, 12.20159247266940, 13.07269285130994, 14.01654260985262, 14.27021942482859, 15.19048615198884, 15.84397997450298, 16.33127799441379, 17.11091235687797, 17.60196514703167, 18.03514871090390

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