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 + 4·7-s − 3·8-s + 9-s + 10-s + 6·11-s + 12-s − 6·13-s + 4·14-s − 15-s − 16-s + 18-s − 2·19-s − 20-s − 4·21-s + 6·22-s + 23-s + 3·24-s + 25-s − 6·26-s − 27-s − 4·28-s − 29-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 − 1.06·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s + 0.288·12-s − 1.66·13-s + 1.06·14-s − 0.258·15-s − 1/4·16-s + 0.235·18-s − 0.458·19-s − 0.223·20-s − 0.872·21-s + 1.27·22-s + 0.208·23-s + 0.612·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.755·28-s − 0.185·29-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 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−17.02445378804956, −16.78845356602984, −15.46588594294871, −14.96039574570667, −14.51993789513594, −14.15448058523263, −13.70615196848260, −12.60411077899314, −12.44578061514101, −11.79300710940675, −11.27860437312383, −10.68181695124016, −9.588230491374269, −9.498408706927200, −8.622977552532016, −8.020646433785280, −6.997797429456753, −6.670437928926493, −5.584470775929995, −5.286923656160924, −4.540157021120125, −4.180863761865367, −3.208818223334106, −1.993631537445095, −1.391094519073730, 0, 1.391094519073730, 1.993631537445095, 3.208818223334106, 4.180863761865367, 4.540157021120125, 5.286923656160924, 5.584470775929995, 6.670437928926493, 6.997797429456753, 8.020646433785280, 8.622977552532016, 9.498408706927200, 9.588230491374269, 10.68181695124016, 11.27860437312383, 11.79300710940675, 12.44578061514101, 12.60411077899314, 13.70615196848260, 14.15448058523263, 14.51993789513594, 14.96039574570667, 15.46588594294871, 16.78845356602984, 17.02445378804956

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