Properties

Label 2-21e2-49.27-c0-0-0
Degree $2$
Conductor $441$
Sign $0.820 - 0.572i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.623 + 0.781i)4-s + (0.900 − 0.433i)7-s + (0.846 + 1.75i)13-s + (−0.222 − 0.974i)16-s + 0.867i·19-s + (−0.900 − 0.433i)25-s + (−0.222 + 0.974i)28-s − 1.56i·31-s + (−0.777 − 0.974i)37-s + (−0.400 − 1.75i)43-s + (0.623 − 0.781i)49-s + (−1.90 − 0.433i)52-s + (−1.22 + 0.974i)61-s + (0.900 + 0.433i)64-s − 0.445·67-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)4-s + (0.900 − 0.433i)7-s + (0.846 + 1.75i)13-s + (−0.222 − 0.974i)16-s + 0.867i·19-s + (−0.900 − 0.433i)25-s + (−0.222 + 0.974i)28-s − 1.56i·31-s + (−0.777 − 0.974i)37-s + (−0.400 − 1.75i)43-s + (0.623 − 0.781i)49-s + (−1.90 − 0.433i)52-s + (−1.22 + 0.974i)61-s + (0.900 + 0.433i)64-s − 0.445·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 - 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 - 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.820 - 0.572i$
Analytic conductor: \(0.220087\)
Root analytic conductor: \(0.469135\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (370, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :0),\ 0.820 - 0.572i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8199486964\)
\(L(\frac12)\) \(\approx\) \(0.8199486964\)
\(L(1)\) 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 \( 1 + (-0.900 + 0.433i)T \)
good2 \( 1 + (0.623 - 0.781i)T^{2} \)
5 \( 1 + (0.900 + 0.433i)T^{2} \)
11 \( 1 + (0.623 - 0.781i)T^{2} \)
13 \( 1 + (-0.846 - 1.75i)T + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + (0.222 - 0.974i)T^{2} \)
19 \( 1 - 0.867iT - T^{2} \)
23 \( 1 + (-0.222 - 0.974i)T^{2} \)
29 \( 1 + (-0.222 + 0.974i)T^{2} \)
31 \( 1 + 1.56iT - T^{2} \)
37 \( 1 + (0.777 + 0.974i)T + (-0.222 + 0.974i)T^{2} \)
41 \( 1 + (0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.400 + 1.75i)T + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.623 + 0.781i)T^{2} \)
53 \( 1 + (-0.222 - 0.974i)T^{2} \)
59 \( 1 + (0.900 - 0.433i)T^{2} \)
61 \( 1 + (1.22 - 0.974i)T + (0.222 - 0.974i)T^{2} \)
67 \( 1 + 0.445T + T^{2} \)
71 \( 1 + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (0.678 - 1.40i)T + (-0.623 - 0.781i)T^{2} \)
79 \( 1 - 1.24T + T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.623 - 0.781i)T^{2} \)
97 \( 1 + 0.867iT - T^{2} \)
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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

−11.64364771649385788898897980470, −10.60004763095687692657772975245, −9.449664008636115201962183406063, −8.661773684538616129423888503632, −7.87604119782695562181828581310, −6.98701937484244941042677298668, −5.67292758131999990757562775390, −4.29872827720491438378431383270, −3.86018323134596211897653580691, −1.93272760017737609322555567365, 1.41866505169058123650939135130, 3.21070083437703102155310474852, 4.74069340594935601604084111127, 5.40500684954348814411753277659, 6.34730658325799405366575058053, 7.85414118753343608850092718838, 8.535151236509782505486619123309, 9.421416255214919497228291777971, 10.50935424216346858335191192172, 10.99291433140716464507877548440

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