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 + 7-s − 8-s + 9-s − 10-s + 2·12-s + 6·13-s − 14-s + 2·15-s + 16-s − 18-s + 7·19-s + 20-s + 2·21-s + 3·23-s − 2·24-s + 25-s − 6·26-s − 4·27-s + 28-s − 6·29-s − 2·30-s − 32-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 + 0.577·12-s + 1.66·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.235·18-s + 1.60·19-s + 0.223·20-s + 0.436·21-s + 0.625·23-s − 0.408·24-s + 1/5·25-s − 1.17·26-s − 0.769·27-s + 0.188·28-s − 1.11·29-s − 0.365·30-s − 0.176·32-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 \( 1 - 2 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−13.86413961889443, −13.55755819072831, −13.35500658075735, −12.59128652668843, −11.91395843037979, −11.35609873428926, −11.12199114728807, −10.43989288867972, −9.898008432070546, −9.401974714820918, −9.049022428898150, −8.458010404601487, −8.253472470092796, −7.661788283251468, −7.066867521351512, −6.603514746558782, −5.886463109199868, −5.356577129976583, −4.863567242636240, −3.670504131313223, −3.531420308523835, −2.970996822812250, −2.189611127405338, −1.485299368732042, −1.262533610086548, 0, 1.262533610086548, 1.485299368732042, 2.189611127405338, 2.970996822812250, 3.531420308523835, 3.670504131313223, 4.863567242636240, 5.356577129976583, 5.886463109199868, 6.603514746558782, 7.066867521351512, 7.661788283251468, 8.253472470092796, 8.458010404601487, 9.049022428898150, 9.401974714820918, 9.898008432070546, 10.43989288867972, 11.12199114728807, 11.35609873428926, 11.91395843037979, 12.59128652668843, 13.35500658075735, 13.55755819072831, 13.86413961889443

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