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 − 4·7-s + 8-s + 9-s + 10-s + 4·11-s + 2·12-s − 2·13-s − 4·14-s + 2·15-s + 16-s − 2·17-s + 18-s + 4·19-s + 20-s − 8·21-s + 4·22-s − 4·23-s + 2·24-s + 25-s − 2·26-s − 4·27-s − 4·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.577·12-s − 0.554·13-s − 1.06·14-s + 0.516·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 1.74·21-s + 0.852·22-s − 0.834·23-s + 0.408·24-s + 1/5·25-s − 0.392·26-s − 0.769·27-s − 0.755·28-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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.90809076862174, −13.73156988680064, −13.06421308081998, −12.71351352448939, −12.20380001808805, −11.68411205320039, −11.24824214262590, −10.25930751922463, −10.05458663221576, −9.443427512213674, −9.173557346685241, −8.646760220669131, −8.009490421005505, −7.282291514794970, −6.958575842904407, −6.377821806724283, −5.893042152994868, −5.457307304189248, −4.316242218204405, −4.258046149197597, −3.278089361696878, −3.144154452746842, −2.525328372926191, −1.898081828591398, −1.114916025005716, 0, 1.114916025005716, 1.898081828591398, 2.525328372926191, 3.144154452746842, 3.278089361696878, 4.258046149197597, 4.316242218204405, 5.457307304189248, 5.893042152994868, 6.377821806724283, 6.958575842904407, 7.282291514794970, 8.009490421005505, 8.646760220669131, 9.173557346685241, 9.443427512213674, 10.05458663221576, 10.25930751922463, 11.24824214262590, 11.68411205320039, 12.20380001808805, 12.71351352448939, 13.06421308081998, 13.73156988680064, 13.90809076862174

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