Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 73 \cdot 137 $
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 + 2·3-s + 4-s − 5-s − 2·6-s − 2·7-s − 8-s + 9-s + 10-s − 2·11-s + 2·12-s − 2·13-s + 2·14-s − 2·15-s + 16-s − 6·17-s − 18-s − 4·19-s − 20-s − 4·21-s + 2·22-s − 6·23-s − 2·24-s + 25-s + 2·26-s − 4·27-s − 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.577·12-s − 0.554·13-s + 0.534·14-s − 0.516·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.872·21-s + 0.426·22-s − 1.25·23-s − 0.408·24-s + 1/5·25-s + 0.392·26-s − 0.769·27-s − 0.377·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(100010\)    =    \(2 \cdot 5 \cdot 73 \cdot 137\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{100010} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 100010,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;73,\;137\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;73,\;137\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
5 \( 1 + T \)
73 \( 1 - T \)
137 \( 1 - T \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.93178954082821, −13.53826391557285, −13.01867385776230, −12.69783746860416, −11.92343091784212, −11.59285192069050, −10.94864476316459, −10.33678396187354, −10.04300365818705, −9.406350210780615, −9.037431968781588, −8.437694470471743, −8.189662685452511, −7.667171950022797, −7.075777597680356, −6.576560727142735, −6.095743290200689, −5.332951875596424, −4.564805261680953, −3.923837660575088, −3.546606345692867, −2.657532633930067, −2.361687168327502, −1.943227818490167, −0.6393430937637406, 0, 0.6393430937637406, 1.943227818490167, 2.361687168327502, 2.657532633930067, 3.546606345692867, 3.923837660575088, 4.564805261680953, 5.332951875596424, 6.095743290200689, 6.576560727142735, 7.075777597680356, 7.667171950022797, 8.189662685452511, 8.437694470471743, 9.037431968781588, 9.406350210780615, 10.04300365818705, 10.33678396187354, 10.94864476316459, 11.59285192069050, 11.92343091784212, 12.69783746860416, 13.01867385776230, 13.53826391557285, 13.93178954082821

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