↑

Properties

Label 2-12221-1.1-c1-0-2
Degree $2$
Conductor $12221$
Sign $-1$
Analytic cond. $97.5851$
Root an. cond. $9.87852$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·4-s − 5-s + 2·7-s + 9-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 − 2·35-s − 2·36-s − 2·37-s + 2·39-s − 8·41-s + 8·43-s − 45-s + 7·47-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s − 0.447·5-s + 0.755·7-s + 1/3·9-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 − 0.338·35-s − 1/3·36-s − 0.328·37-s + 0.320·39-s − 1.24·41-s + 1.21·43-s − 0.149·45-s + 1.02·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12221 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12221 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12221\)    =    \(11^{2} \cdot 101\)
Sign: $-1$
Analytic conductor: \(97.5851\)
Root analytic conductor: \(9.87852\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 12221,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
101 \( 1 - T \)
good2 \( 1 + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.85980790520647, −16.06597881470185, −15.63906880237070, −14.90789931195245, −14.30183817659289, −13.81387390742372, −13.26701081495265, −12.31863492171253, −12.19448667046163, −11.48556520347371, −10.89922113262013, −10.52197915740416, −9.604963054543405, −9.124813249388043, −8.442095246972987, −7.748783150728753, −7.271233914122942, −6.336287353983028, −5.662227775480124, −5.051900116833679, −4.708987726243881, −3.911871784751907, −3.142238064239636, −1.858660979153620, −0.8456385846119239, 0, 0.8456385846119239, 1.858660979153620, 3.142238064239636, 3.911871784751907, 4.708987726243881, 5.051900116833679, 5.662227775480124, 6.336287353983028, 7.271233914122942, 7.748783150728753, 8.442095246972987, 9.124813249388043, 9.604963054543405, 10.52197915740416, 10.89922113262013, 11.48556520347371, 12.19448667046163, 12.31863492171253, 13.26701081495265, 13.81387390742372, 14.30183817659289, 14.90789931195245, 15.63906880237070, 16.06597881470185, 16.85980790520647

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