Properties

Degree 2
Conductor 101
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·4-s − 5-s − 2·7-s + 9-s − 2·11-s + 4·12-s + 13-s + 2·15-s + 4·16-s + 3·17-s − 5·19-s + 2·20-s + 4·21-s + 23-s − 4·25-s + 4·27-s + 4·28-s − 4·29-s − 9·31-s + 4·33-s + 2·35-s − 2·36-s − 2·37-s − 2·39-s + 8·41-s − 8·43-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 1.15·12-s + 0.277·13-s + 0.516·15-s + 16-s + 0.727·17-s − 1.14·19-s + 0.447·20-s + 0.872·21-s + 0.208·23-s − 4/5·25-s + 0.769·27-s + 0.755·28-s − 0.742·29-s − 1.61·31-s + 0.696·33-s + 0.338·35-s − 1/3·36-s − 0.328·37-s − 0.320·39-s + 1.24·41-s − 1.21·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(101\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{101} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 101,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 101$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 101$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad101 \( 1 + T \)
good2 \( 1 + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−19.85906713457606, −18.85694039545399, −18.24606910263464, −17.17010274227894, −16.57421300714814, −15.50733332835016, −14.30115955003167, −13.00508217019689, −12.46381898373739, −11.18987953654912, −10.24202176919573, −9.060317898736740, −7.741879349102782, −6.160551164578111, −5.173304987352839, −3.729334683400909, 0, 3.729334683400909, 5.173304987352839, 6.160551164578111, 7.741879349102782, 9.060317898736740, 10.24202176919573, 11.18987953654912, 12.46381898373739, 13.00508217019689, 14.30115955003167, 15.50733332835016, 16.57421300714814, 17.17010274227894, 18.24606910263464, 18.85694039545399, 19.85906713457606

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