Properties

Degree $2$
Conductor $80223$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 4-s − 2·5-s − 6-s − 3·8-s + 9-s − 2·10-s + 12-s − 13-s + 2·15-s − 16-s − 17-s + 18-s + 4·19-s + 2·20-s + 3·24-s − 25-s − 26-s − 27-s + 2·29-s + 2·30-s − 8·31-s + 5·32-s − 34-s − 36-s − 2·37-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.632·10-s + 0.288·12-s − 0.277·13-s + 0.516·15-s − 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.612·24-s − 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.371·29-s + 0.365·30-s − 1.43·31-s + 0.883·32-s − 0.171·34-s − 1/6·36-s − 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80223\)    =    \(3 \cdot 11^{2} \cdot 13 \cdot 17\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{80223} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 80223,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5974916754\)
\(L(\frac12)\) \(\approx\) \(0.5974916754\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
11 \( 1 \)
13 \( 1 + T \)
17 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.85213337040087, −13.64311191551793, −12.85432306327749, −12.34591112134823, −12.29859911365268, −11.54576100915024, −11.20039645897983, −10.67546900803568, −9.969566104722212, −9.413067592558907, −9.067464235564383, −8.368636031462496, −7.729004013825360, −7.434109626540978, −6.674124285548644, −6.153506639609774, −5.517239433724197, −5.088898260736896, −4.565066131373711, −4.018865178446543, −3.496052280193142, −3.003750485889008, −2.073187345113229, −1.134986205714987, −0.2597210392578932, 0.2597210392578932, 1.134986205714987, 2.073187345113229, 3.003750485889008, 3.496052280193142, 4.018865178446543, 4.565066131373711, 5.088898260736896, 5.517239433724197, 6.153506639609774, 6.674124285548644, 7.434109626540978, 7.729004013825360, 8.368636031462496, 9.067464235564383, 9.413067592558907, 9.969566104722212, 10.67546900803568, 11.20039645897983, 11.54576100915024, 12.29859911365268, 12.34591112134823, 12.85432306327749, 13.64311191551793, 13.85213337040087

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