↑

Properties

Label 2-105e2-1.1-c1-0-21
Degree $2$
Conductor $11025$
Sign $-1$
Analytic cond. $88.0350$
Root an. cond. $9.38270$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 11-s − 3·13-s − 4·16-s − 3·17-s + 6·19-s + 2·22-s − 4·23-s + 6·26-s + 29-s + 6·31-s + 8·32-s + 6·34-s − 12·38-s − 6·41-s + 6·43-s − 2·44-s + 8·46-s − 9·47-s − 6·52-s − 10·53-s − 2·58-s + 6·59-s − 12·62-s − 8·64-s + 14·67-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.301·11-s − 0.832·13-s − 16-s − 0.727·17-s + 1.37·19-s + 0.426·22-s − 0.834·23-s + 1.17·26-s + 0.185·29-s + 1.07·31-s + 1.41·32-s + 1.02·34-s − 1.94·38-s − 0.937·41-s + 0.914·43-s − 0.301·44-s + 1.17·46-s − 1.31·47-s − 0.832·52-s − 1.37·53-s − 0.262·58-s + 0.781·59-s − 1.52·62-s − 64-s + 1.71·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11025\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(88.0350\)
Root analytic conductor: \(9.38270\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 11025,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 15 T + p T^{2} \)
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

−17.08038111859992, −16.18914897117736, −15.85978844009378, −15.43449513835802, −14.47092260480147, −14.00012972714077, −13.38289086192968, −12.70983800896383, −11.90761007539713, −11.48736449800086, −10.85461009338794, −10.05153356072472, −9.838426316758612, −9.247178719559399, −8.493117232463389, −7.983628187522681, −7.474986339172224, −6.804158228802259, −6.183750937074146, −5.101311624043889, −4.677890835444351, −3.613457968809653, −2.641770617912970, −1.961164488659942, −0.9655816077082292, 0, 0.9655816077082292, 1.961164488659942, 2.641770617912970, 3.613457968809653, 4.677890835444351, 5.101311624043889, 6.183750937074146, 6.804158228802259, 7.474986339172224, 7.983628187522681, 8.493117232463389, 9.247178719559399, 9.838426316758612, 10.05153356072472, 10.85461009338794, 11.48736449800086, 11.90761007539713, 12.70983800896383, 13.38289086192968, 14.00012972714077, 14.47092260480147, 15.43449513835802, 15.85978844009378, 16.18914897117736, 17.08038111859992

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