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-s + 3-s − 4-s − 5-s + 6-s − 3·8-s + 9-s − 10-s + 2·11-s − 12-s + 2·13-s − 15-s − 16-s − 4·17-s + 18-s + 20-s + 2·22-s − 23-s − 3·24-s + 25-s + 2·26-s + 27-s + 29-s − 30-s − 2·31-s + 5·32-s + 2·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s − 0.288·12-s + 0.554·13-s − 0.258·15-s − 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.223·20-s + 0.426·22-s − 0.208·23-s − 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.185·29-s − 0.182·30-s − 0.359·31-s + 0.883·32-s + 0.348·33-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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
5 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−16.97253854161408, −16.01179480526197, −15.74757207708826, −15.05369772001261, −14.47063841160399, −14.19631790009649, −13.32642379498414, −13.17363669767037, −12.38409738061395, −11.89770852496971, −11.15250393104525, −10.63733981883944, −9.492142936172275, −9.367147370219083, −8.551250666566472, −8.126507112848970, −7.322461683746430, −6.419173156585503, −6.080104760162119, −4.951237660074223, −4.510893000303053, −3.776916385125830, −3.310383577653069, −2.393326952489312, −1.287333937066845, 0, 1.287333937066845, 2.393326952489312, 3.310383577653069, 3.776916385125830, 4.510893000303053, 4.951237660074223, 6.080104760162119, 6.419173156585503, 7.322461683746430, 8.126507112848970, 8.551250666566472, 9.367147370219083, 9.492142936172275, 10.63733981883944, 11.15250393104525, 11.89770852496971, 12.38409738061395, 13.17363669767037, 13.32642379498414, 14.19631790009649, 14.47063841160399, 15.05369772001261, 15.74757207708826, 16.01179480526197, 16.97253854161408

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