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Properties

Label 2-21e2-21.2-c0-0-0
Degree $2$
Conductor $441$
Sign $0.469 + 0.882i$
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  + (−1.22 − 0.707i)2-s + (0.499 + 0.866i)4-s + (1.22 − 0.707i)11-s + (0.499 − 0.866i)16-s − 2·22-s + (1.22 + 0.707i)23-s + (−0.5 − 0.866i)25-s − 1.41i·29-s + (−1.22 + 0.707i)32-s + (1.22 + 0.707i)44-s + (−0.999 − 1.73i)46-s + 1.41i·50-s + (−1.22 + 0.707i)53-s + (−1.00 + 1.73i)58-s + 0.999·64-s + ⋯
L(s)  = 1  + (−1.22 − 0.707i)2-s + (0.499 + 0.866i)4-s + (1.22 − 0.707i)11-s + (0.499 − 0.866i)16-s − 2·22-s + (1.22 + 0.707i)23-s + (−0.5 − 0.866i)25-s − 1.41i·29-s + (−1.22 + 0.707i)32-s + (1.22 + 0.707i)44-s + (−0.999 − 1.73i)46-s + 1.41i·50-s + (−1.22 + 0.707i)53-s + (−1.00 + 1.73i)58-s + 0.999·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.469 + 0.882i$
Analytic conductor: \(0.220087\)
Root analytic conductor: \(0.469135\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (422, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :0),\ 0.469 + 0.882i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4946526962\)
\(L(\frac12)\) \(\approx\) \(0.4946526962\)
\(L(1)\) 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 \( 1 \)
good2 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−11.30002800260642342599704048754, −10.13444151591728093797579546615, −9.437568863336295526043237869777, −8.697862528249486234071116720372, −7.889531821328608948275817454481, −6.73258232742497397747311064116, −5.58395740293661999262320915742, −4.03905757834259701265667766827, −2.71155808384978503277722180903, −1.22447069934551854485032977313, 1.51270748961077959396563896666, 3.53857529263476151221789652631, 4.91076093647297368638248374155, 6.36194329576796973093361987275, 6.98762824746807386839413737328, 7.84273445823638068332700752822, 8.991725428697495458276048686635, 9.303193161721400397952824269371, 10.34348904407705027774282178700, 11.21850204661093033399877615259

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