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 + 4·11-s + 12-s − 2·13-s − 15-s − 16-s − 6·17-s + 18-s + 4·19-s − 20-s + 4·22-s − 23-s + 3·24-s + 25-s − 2·26-s − 27-s + 29-s − 30-s + 4·31-s + 5·32-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 + 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.258·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.852·22-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 + 0.718·31-s + 0.883·32-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 $F_p$ 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−17.17718612691170, −16.30636475088869, −15.73588379599894, −15.09616656410795, −14.50618449612901, −14.01570979062440, −13.46538500862401, −13.00211296240947, −12.27194063929149, −11.80240693551841, −11.37018780481208, −10.49094406420653, −9.768763226507564, −9.338643346579857, −8.768821891142581, −7.992315866950088, −6.946005974467450, −6.489899452271880, −5.962355840346338, −5.032412579610391, −4.749854360805160, −3.949153321139294, −3.229960703715178, −2.218764488273508, −1.180916365052916, 0, 1.180916365052916, 2.218764488273508, 3.229960703715178, 3.949153321139294, 4.749854360805160, 5.032412579610391, 5.962355840346338, 6.489899452271880, 6.946005974467450, 7.992315866950088, 8.768821891142581, 9.338643346579857, 9.768763226507564, 10.49094406420653, 11.37018780481208, 11.80240693551841, 12.27194063929149, 13.00211296240947, 13.46538500862401, 14.01570979062440, 14.50618449612901, 15.09616656410795, 15.73588379599894, 16.30636475088869, 17.17718612691170

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