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 + 3·3-s + 4-s + 5-s + 3·6-s + 3·7-s + 8-s + 6·9-s + 10-s + 11-s + 3·12-s − 6·13-s + 3·14-s + 3·15-s + 16-s − 2·17-s + 6·18-s − 8·19-s + 20-s + 9·21-s + 22-s + 8·23-s + 3·24-s + 25-s − 6·26-s + 9·27-s + 3·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 1/2·4-s + 0.447·5-s + 1.22·6-s + 1.13·7-s + 0.353·8-s + 2·9-s + 0.316·10-s + 0.301·11-s + 0.866·12-s − 1.66·13-s + 0.801·14-s + 0.774·15-s + 1/4·16-s − 0.485·17-s + 1.41·18-s − 1.83·19-s + 0.223·20-s + 1.96·21-s + 0.213·22-s + 1.66·23-s + 0.612·24-s + 1/5·25-s − 1.17·26-s + 1.73·27-s + 0.566·28-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - T \)
73 \( 1 + T \)
137 \( 1 + T \)
good3 \( 1 - p T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−14.19676195478821, −13.58790898342947, −13.05287034884728, −12.76056634277165, −12.35036319594618, −11.55640924923346, −11.07941164001386, −10.48894548190499, −10.11116370053389, −9.340865289053838, −8.979353760275522, −8.591700754568021, −8.020455650531728, −7.433248513061049, −7.055740945131127, −6.628154634731643, −5.750579244986391, −4.954413376174514, −4.671350341007062, −4.290532100477362, −3.324885871905414, −3.072403878250258, −2.248939507268436, −1.831438718402718, −1.571437694287410, 0, 1.571437694287410, 1.831438718402718, 2.248939507268436, 3.072403878250258, 3.324885871905414, 4.290532100477362, 4.671350341007062, 4.954413376174514, 5.750579244986391, 6.628154634731643, 7.055740945131127, 7.433248513061049, 8.020455650531728, 8.591700754568021, 8.979353760275522, 9.340865289053838, 10.11116370053389, 10.48894548190499, 11.07941164001386, 11.55640924923346, 12.35036319594618, 12.76056634277165, 13.05287034884728, 13.58790898342947, 14.19676195478821

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