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Properties

Label 2-21e2-7.4-c1-0-8
Degree $2$
Conductor $441$
Sign $0.266 + 0.963i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.500 − 0.866i)4-s + (1 + 1.73i)5-s − 3·8-s + (0.999 − 1.73i)10-s + (2 − 3.46i)11-s + 2·13-s + (0.500 + 0.866i)16-s + (3 − 5.19i)17-s + (2 + 3.46i)19-s + 2·20-s − 3.99·22-s + (0.500 − 0.866i)25-s + (−1 − 1.73i)26-s + 2·29-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.250 − 0.433i)4-s + (0.447 + 0.774i)5-s − 1.06·8-s + (0.316 − 0.547i)10-s + (0.603 − 1.04i)11-s + 0.554·13-s + (0.125 + 0.216i)16-s + (0.727 − 1.26i)17-s + (0.458 + 0.794i)19-s + 0.447·20-s − 0.852·22-s + (0.100 − 0.173i)25-s + (−0.196 − 0.339i)26-s + 0.371·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.266 + 0.963i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08560 - 0.825880i\)
\(L(\frac12)\) \(\approx\) \(1.08560 - 0.825880i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (3 - 5.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-7 - 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 18T + 97T^{2} \)
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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

−10.89860858961831135845621597150, −10.12863263573002084837923822478, −9.402543631248413595190775792406, −8.448713088067621153396401594024, −7.09363382810483094350572534532, −6.18650228057432821400645670841, −5.45336470547072959109809118021, −3.55722881914239991028849963167, −2.63453838237042863909515533895, −1.11564626932803369620485645805, 1.60812344594404046507223451442, 3.31773101016116623867932707704, 4.63372453164820815437507166258, 5.84256514333728356367828773146, 6.71877183662010098343453382951, 7.66969197391076211224592416394, 8.604500326453477787828110058279, 9.245938166960791646531176792884, 10.17990254695275946028945551699, 11.42371735861110295501405281529

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