Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 23 \cdot 29 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3-s + 2·4-s − 5-s − 2·6-s + 7-s + 9-s − 2·10-s − 2·11-s − 2·12-s + 2·14-s + 15-s − 4·16-s + 3·17-s + 2·18-s + 5·19-s − 2·20-s − 21-s − 4·22-s + 23-s + 25-s − 27-s + 2·28-s + 29-s + 2·30-s − 4·31-s − 8·32-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s + 0.377·7-s + 1/3·9-s − 0.632·10-s − 0.603·11-s − 0.577·12-s + 0.534·14-s + 0.258·15-s − 16-s + 0.727·17-s + 0.471·18-s + 1.14·19-s − 0.447·20-s − 0.218·21-s − 0.852·22-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.377·28-s + 0.185·29-s + 0.365·30-s − 0.718·31-s − 1.41·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10005\)    =    \(3 \cdot 5 \cdot 23 \cdot 29\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{10005} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 10005,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;23,\;29\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;23,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 - T \)
good2 \( 1 - p T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−16.79025186014036, −15.95677439850830, −15.86243487432184, −15.07921429867695, −14.64669558764262, −13.91419041923949, −13.58219555213783, −12.81758723532614, −12.29740166028256, −11.92043248483400, −11.29287481596666, −10.81540532756453, −10.02410170799300, −9.353127250940880, −8.435229388177375, −7.796780697148437, −7.051750121822555, −6.546307663788831, −5.570797241199663, −5.194549161708319, −4.809737146618708, −3.734191969042389, −3.406857448368338, −2.438025953751510, −1.332242146878808, 0, 1.332242146878808, 2.438025953751510, 3.406857448368338, 3.734191969042389, 4.809737146618708, 5.194549161708319, 5.570797241199663, 6.546307663788831, 7.051750121822555, 7.796780697148437, 8.435229388177375, 9.353127250940880, 10.02410170799300, 10.81540532756453, 11.29287481596666, 11.92043248483400, 12.29740166028256, 12.81758723532614, 13.58219555213783, 13.91419041923949, 14.64669558764262, 15.07921429867695, 15.86243487432184, 15.95677439850830, 16.79025186014036

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