Properties

Degree 2
Conductor $ 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 − 4-s − 4·5-s − 4·7-s − 3·8-s − 3·9-s − 4·10-s + 4·11-s − 4·14-s − 16-s + 2·17-s − 3·18-s − 4·19-s + 4·20-s + 4·22-s + 4·23-s + 11·25-s + 4·28-s − 4·29-s − 2·31-s + 5·32-s + 2·34-s + 16·35-s + 3·36-s + 6·37-s − 4·38-s + 12·40-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.78·5-s − 1.51·7-s − 1.06·8-s − 9-s − 1.26·10-s + 1.20·11-s − 1.06·14-s − 1/4·16-s + 0.485·17-s − 0.707·18-s − 0.917·19-s + 0.894·20-s + 0.852·22-s + 0.834·23-s + 11/5·25-s + 0.755·28-s − 0.742·29-s − 0.359·31-s + 0.883·32-s + 0.342·34-s + 2.70·35-s + 1/2·36-s + 0.986·37-s − 0.648·38-s + 1.89·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10001\)    =    \(73 \cdot 137\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{10001} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 10001,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{73,\;137\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{73,\;137\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad73 \( 1 - T \)
137 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
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

−16.83818430649483, −16.36596747611176, −15.60478010452076, −15.15281101720038, −14.57421708887065, −14.28379845623159, −13.30802846135986, −12.86169562443263, −12.24172147358369, −12.01212029107784, −11.19099123788464, −10.81295475969948, −9.563796200572135, −9.252289626241779, −8.647133885305995, −8.019494782556084, −7.221127011782987, −6.479878083322923, −6.042081009666289, −5.154411930752986, −4.254429892912670, −3.824786579941877, −3.309544997528499, −2.720165075912424, −0.7738720728652220, 0, 0.7738720728652220, 2.720165075912424, 3.309544997528499, 3.824786579941877, 4.254429892912670, 5.154411930752986, 6.042081009666289, 6.479878083322923, 7.221127011782987, 8.019494782556084, 8.647133885305995, 9.252289626241779, 9.563796200572135, 10.81295475969948, 11.19099123788464, 12.01212029107784, 12.24172147358369, 12.86169562443263, 13.30802846135986, 14.28379845623159, 14.57421708887065, 15.15281101720038, 15.60478010452076, 16.36596747611176, 16.83818430649483

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