# Properties

 Label 2-21e2-9.4-c1-0-30 Degree $2$ Conductor $441$ Sign $0.722 + 0.690i$ 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 + (−0.0341 − 0.0592i)2-s + (1.69 + 0.355i)3-s + (0.997 − 1.72i)4-s + (1.33 − 2.30i)5-s + (−0.0368 − 0.112i)6-s − 0.273·8-s + (2.74 + 1.20i)9-s − 0.182·10-s + (0.799 + 1.38i)11-s + (2.30 − 2.57i)12-s + (−2.62 + 4.54i)13-s + (3.07 − 3.43i)15-s + (−1.98 − 3.43i)16-s − 6.54·17-s + (−0.0225 − 0.203i)18-s + 1.90·19-s + ⋯
 L(s)  = 1 + (−0.0241 − 0.0418i)2-s + (0.978 + 0.205i)3-s + (0.498 − 0.864i)4-s + (0.595 − 1.03i)5-s + (−0.0150 − 0.0459i)6-s − 0.0965·8-s + (0.915 + 0.401i)9-s − 0.0575·10-s + (0.241 + 0.417i)11-s + (0.665 − 0.743i)12-s + (−0.728 + 1.26i)13-s + (0.794 − 0.887i)15-s + (−0.496 − 0.859i)16-s − 1.58·17-s + (−0.00530 − 0.0480i)18-s + 0.436·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $0.722 + 0.690i$ Analytic conductor: $$3.52140$$ Root analytic conductor: $$1.87654$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{441} (148, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 441,\ (\ :1/2),\ 0.722 + 0.690i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.05724 - 0.824913i$$ $$L(\frac12)$$ $$\approx$$ $$2.05724 - 0.824913i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-1.69 - 0.355i)T$$
7 $$1$$
good2 $$1 + (0.0341 + 0.0592i)T + (-1 + 1.73i)T^{2}$$
5 $$1 + (-1.33 + 2.30i)T + (-2.5 - 4.33i)T^{2}$$
11 $$1 + (-0.799 - 1.38i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 + (2.62 - 4.54i)T + (-6.5 - 11.2i)T^{2}$$
17 $$1 + 6.54T + 17T^{2}$$
19 $$1 - 1.90T + 19T^{2}$$
23 $$1 + (-1.53 + 2.65i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (3.19 + 5.53i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + (3.35 - 5.81i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 - 4.22T + 37T^{2}$$
41 $$1 + (3.69 - 6.40i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (-5.63 - 9.75i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (1.89 + 3.29i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 - 8.89T + 53T^{2}$$
59 $$1 + (-5.44 + 9.43i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (1.35 + 2.35i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-1.66 + 2.87i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + 12.3T + 71T^{2}$$
73 $$1 - 2.19T + 73T^{2}$$
79 $$1 + (0.406 + 0.704i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + (3.41 + 5.92i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 - 0.470T + 89T^{2}$$
97 $$1 + (-2.57 - 4.46i)T + (-48.5 + 84.0i)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

−10.90516858970310816601254447929, −9.674684056818900422512331606517, −9.426543296171161091055704654978, −8.608044647993126843988323816793, −7.24069915460356069516558550976, −6.43228621412110146459335141518, −5.00481274697328436606024748911, −4.37091838195637321145720315633, −2.43106263021967697165113718218, −1.58963681226248689487931915340, 2.24065871890587052691482917074, 2.92955848042129448054602103698, 3.94285167560328458244892788218, 5.76058430310140454098248358587, 7.03958652092189889453729992080, 7.31687864390236402685075544673, 8.484139283662917271394695880728, 9.277291828909427931287232269779, 10.39094411664134690012437632230, 11.10225352618162136777478697846