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  + 3-s − 2·4-s − 5-s + 2·7-s + 9-s − 4·11-s − 2·12-s − 5·13-s − 15-s + 4·16-s + 7·17-s − 5·19-s + 2·20-s + 2·21-s + 23-s + 25-s + 27-s − 4·28-s − 29-s + 8·31-s − 4·33-s − 2·35-s − 2·36-s − 37-s − 5·39-s + 4·41-s − 43-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.577·12-s − 1.38·13-s − 0.258·15-s + 16-s + 1.69·17-s − 1.14·19-s + 0.447·20-s + 0.436·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.755·28-s − 0.185·29-s + 1.43·31-s − 0.696·33-s − 0.338·35-s − 1/3·36-s − 0.164·37-s − 0.800·39-s + 0.624·41-s − 0.152·43-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 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 13 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−17.12007203340007, −16.43982388267547, −15.55322962115916, −15.07506927365860, −14.62270893739293, −14.09938565181892, −13.63293446934501, −12.80426515770958, −12.36672320432773, −12.02162085652916, −10.77777805279746, −10.56756210753832, −9.586607209339136, −9.478274215242557, −8.294504890771797, −7.995153218793792, −7.769028127935665, −6.817299588396222, −5.714835252724412, −4.997800812886961, −4.683146342159354, −3.859671729486485, −3.005818808665905, −2.319542720153191, −1.113624279833670, 0, 1.113624279833670, 2.319542720153191, 3.005818808665905, 3.859671729486485, 4.683146342159354, 4.997800812886961, 5.714835252724412, 6.817299588396222, 7.769028127935665, 7.995153218793792, 8.294504890771797, 9.478274215242557, 9.586607209339136, 10.56756210753832, 10.77777805279746, 12.02162085652916, 12.36672320432773, 12.80426515770958, 13.63293446934501, 14.09938565181892, 14.62270893739293, 15.07506927365860, 15.55322962115916, 16.43982388267547, 17.12007203340007

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