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 − 4·19-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 − 0.917·19-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 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−17.21141987171823, −16.58095849744622, −16.09597925602844, −15.23274429746703, −14.66955859545616, −13.84023657305319, −13.64430650843006, −12.71098191382802, −12.17220543371220, −12.00914051529595, −10.79295640910548, −10.27978725019290, −9.788495411727593, −9.316525768148544, −8.950752320767353, −8.046253355025427, −7.127975024929143, −6.660696270173835, −6.208994220590045, −5.301668255972955, −4.463676274078418, −3.986236478199587, −2.956199407473549, −1.943849074359578, −0.8948665291123890, 0, 0.8948665291123890, 1.943849074359578, 2.956199407473549, 3.986236478199587, 4.463676274078418, 5.301668255972955, 6.208994220590045, 6.660696270173835, 7.127975024929143, 8.046253355025427, 8.950752320767353, 9.316525768148544, 9.788495411727593, 10.27978725019290, 10.79295640910548, 12.00914051529595, 12.17220543371220, 12.71098191382802, 13.64430650843006, 13.84023657305319, 14.66955859545616, 15.23274429746703, 16.09597925602844, 16.58095849744622, 17.21141987171823

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