↑

Properties

Label 2-21e2-49.41-c0-0-0
Degree $2$
Conductor $441$
Sign $0.545 + 0.838i$
Analytic cond. $0.220087$
Root an. cond. $0.469135$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.222 − 0.974i)4-s + (−0.623 − 0.781i)7-s + (0.678 − 0.541i)13-s + (−0.900 − 0.433i)16-s + 1.56i·19-s + (0.623 − 0.781i)25-s + (−0.900 + 0.433i)28-s + 1.94i·31-s + (−0.0990 − 0.433i)37-s + (1.12 + 0.541i)43-s + (−0.222 + 0.974i)49-s + (−0.376 − 0.781i)52-s + (−1.90 + 0.433i)61-s + (−0.623 + 0.781i)64-s − 1.80·67-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)4-s + (−0.623 − 0.781i)7-s + (0.678 − 0.541i)13-s + (−0.900 − 0.433i)16-s + 1.56i·19-s + (0.623 − 0.781i)25-s + (−0.900 + 0.433i)28-s + 1.94i·31-s + (−0.0990 − 0.433i)37-s + (1.12 + 0.541i)43-s + (−0.222 + 0.974i)49-s + (−0.376 − 0.781i)52-s + (−1.90 + 0.433i)61-s + (−0.623 + 0.781i)64-s − 1.80·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.545 + 0.838i$
Analytic conductor: \(0.220087\)
Root analytic conductor: \(0.469135\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :0),\ 0.545 + 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8489066916\)
\(L(\frac12)\) \(\approx\) \(0.8489066916\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (0.623 + 0.781i)T \)
good2 \( 1 + (-0.222 + 0.974i)T^{2} \)
5 \( 1 + (-0.623 + 0.781i)T^{2} \)
11 \( 1 + (-0.222 + 0.974i)T^{2} \)
13 \( 1 + (-0.678 + 0.541i)T + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (0.900 - 0.433i)T^{2} \)
19 \( 1 - 1.56iT - T^{2} \)
23 \( 1 + (-0.900 - 0.433i)T^{2} \)
29 \( 1 + (-0.900 + 0.433i)T^{2} \)
31 \( 1 - 1.94iT - T^{2} \)
37 \( 1 + (0.0990 + 0.433i)T + (-0.900 + 0.433i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (-1.12 - 0.541i)T + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (0.222 - 0.974i)T^{2} \)
53 \( 1 + (-0.900 - 0.433i)T^{2} \)
59 \( 1 + (-0.623 - 0.781i)T^{2} \)
61 \( 1 + (1.90 - 0.433i)T + (0.900 - 0.433i)T^{2} \)
67 \( 1 + 1.80T + T^{2} \)
71 \( 1 + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (-1.52 - 1.21i)T + (0.222 + 0.974i)T^{2} \)
79 \( 1 + 0.445T + T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.222 + 0.974i)T^{2} \)
97 \( 1 + 1.56iT - T^{2} \)
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

−10.74004825271992339128190944293, −10.51632379758916736520286603781, −9.611917035846751202924724417821, −8.543968610326092762144155808766, −7.37879854559053345863740111281, −6.40750299438881782695695853141, −5.68000845334625560287929794526, −4.38668676347534611002566275752, −3.12611316205290176224731723765, −1.32729449626805128813890952013, 2.37718199879360827484788402084, 3.39025340258085313615061057248, 4.59019364907655445378541667039, 6.00368311096996551872152539710, 6.86249787031418915063432790231, 7.81392504079458448654985461958, 8.979263908649033912625157754504, 9.280927662016163499450746122033, 10.84160407478277154345908583756, 11.53490723769160968443211569253

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