# Properties

 Label 2-21e2-7.2-c1-0-12 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 $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.866 − 1.5i)2-s + (−0.5 − 0.866i)4-s + (1.73 − 3i)5-s + 1.73·8-s + (−3 − 5.19i)10-s + (1.73 + 3i)11-s − 2·13-s + (2.49 − 4.33i)16-s + (−1.73 − 3i)17-s + (−2 + 3.46i)19-s − 3.46·20-s + 6·22-s + (−1.73 + 3i)23-s + (−3.5 − 6.06i)25-s + (−1.73 + 3i)26-s + ⋯
 L(s)  = 1 + (0.612 − 1.06i)2-s + (−0.250 − 0.433i)4-s + (0.774 − 1.34i)5-s + 0.612·8-s + (−0.948 − 1.64i)10-s + (0.522 + 0.904i)11-s − 0.554·13-s + (0.624 − 1.08i)16-s + (−0.420 − 0.727i)17-s + (−0.458 + 0.794i)19-s − 0.774·20-s + 1.27·22-s + (−0.361 + 0.625i)23-s + (−0.700 − 1.21i)25-s + (−0.339 + 0.588i)26-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: $\chi_{441} (226, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 441,\ (\ :1/2),\ -0.266 + 0.963i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.36065 - 1.78855i$$ $$L(\frac12)$$ $$\approx$$ $$1.36065 - 1.78855i$$ $$L(\frac{3}{2})$$ 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 + (-0.866 + 1.5i)T + (-1 - 1.73i)T^{2}$$
5 $$1 + (-1.73 + 3i)T + (-2.5 - 4.33i)T^{2}$$
11 $$1 + (-1.73 - 3i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 + 2T + 13T^{2}$$
17 $$1 + (1.73 + 3i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (1.73 - 3i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 - 10.3T + 41T^{2}$$
43 $$1 + 4T + 43T^{2}$$
47 $$1 + (3.46 - 6i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (3.46 + 6i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (-3.46 - 6i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 - 10.3T + 71T^{2}$$
73 $$1 + (-7 - 12.1i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + 83T^{2}$$
89 $$1 + (-1.73 + 3i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + 14T + 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

−11.07597797889966436151275821958, −9.762108333841028322958995533774, −9.577924777195509527047162479013, −8.276255283095500725974311184833, −7.15493606615762902651882748294, −5.70767997243788889678556460443, −4.76788602342978370105735376727, −4.05027739523580402681731346377, −2.39308240494693348068871152938, −1.42585812511258605735415408409, 2.17410195544381373307158306863, 3.57164751626833891532980491553, 4.89009565842659254257570699128, 6.11017428410712228794791537419, 6.45490217576550286973194696935, 7.30784016852034462052284285322, 8.435306804017051460939841793516, 9.608106847756638524911174073172, 10.71931607229873905240695053482, 11.02090591274987117362807617047