Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 7 \cdot 11 \cdot 13 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s − 5-s + 2·6-s + 7-s − 8-s + 9-s + 10-s + 11-s − 2·12-s + 13-s − 14-s + 2·15-s + 16-s − 18-s + 2·19-s − 20-s − 2·21-s − 22-s − 6·23-s + 2·24-s + 25-s − 26-s + 4·27-s + 28-s − 6·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.577·12-s + 0.277·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.235·18-s + 0.458·19-s − 0.223·20-s − 0.436·21-s − 0.213·22-s − 1.25·23-s + 0.408·24-s + 1/5·25-s − 0.196·26-s + 0.769·27-s + 0.188·28-s − 1.11·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 10010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 10010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10010\)    =    \(2 \cdot 5 \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{10010} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 10010,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6077165723$
$L(\frac12)$  $\approx$  $0.6077165723$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;7,\;11,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−16.70884151650296, −16.26060396360027, −15.72321303515505, −15.11866385038805, −14.38495368340942, −13.89628110373657, −12.92536736626161, −12.31573225186273, −11.85243150159084, −11.27802248797989, −10.93672301937364, −10.33961909242618, −9.531051816438761, −9.019346669262517, −8.182026892917682, −7.679627749503156, −7.026148798399761, −6.251533512973244, −5.753208979239760, −5.097913658790192, −4.195841229052003, −3.506310872153976, −2.359612470572262, −1.404268135002306, −0.4651502728336448, 0.4651502728336448, 1.404268135002306, 2.359612470572262, 3.506310872153976, 4.195841229052003, 5.097913658790192, 5.753208979239760, 6.251533512973244, 7.026148798399761, 7.679627749503156, 8.182026892917682, 9.019346669262517, 9.531051816438761, 10.33961909242618, 10.93672301937364, 11.27802248797989, 11.85243150159084, 12.31573225186273, 12.92536736626161, 13.89628110373657, 14.38495368340942, 15.11866385038805, 15.72321303515505, 16.26060396360027, 16.70884151650296

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