# Properties

 Label 2-420-60.59-c1-0-48 Degree $2$ Conductor $420$ Sign $0.805 + 0.592i$ Analytic cond. $3.35371$ Root an. cond. $1.83131$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.41i·2-s + (−1 − 1.41i)3-s − 2.00·4-s + (2.12 − 0.707i)5-s + (2.00 − 1.41i)6-s + 7-s − 2.82i·8-s + (−1.00 + 2.82i)9-s + (1.00 + 3i)10-s − 4.24·11-s + (2.00 + 2.82i)12-s − 6i·13-s + 1.41i·14-s + (−3.12 − 2.29i)15-s + 4.00·16-s + 4.24·17-s + ⋯
 L(s)  = 1 + 0.999i·2-s + (−0.577 − 0.816i)3-s − 1.00·4-s + (0.948 − 0.316i)5-s + (0.816 − 0.577i)6-s + 0.377·7-s − 1.00i·8-s + (−0.333 + 0.942i)9-s + (0.316 + 0.948i)10-s − 1.27·11-s + (0.577 + 0.816i)12-s − 1.66i·13-s + 0.377i·14-s + (−0.805 − 0.592i)15-s + 1.00·16-s + 1.02·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$420$$    =    $$2^{2} \cdot 3 \cdot 5 \cdot 7$$ Sign: $0.805 + 0.592i$ Analytic conductor: $$3.35371$$ Root analytic conductor: $$1.83131$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{420} (239, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 420,\ (\ :1/2),\ 0.805 + 0.592i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.01983 - 0.334324i$$ $$L(\frac12)$$ $$\approx$$ $$1.01983 - 0.334324i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 1.41iT$$
3 $$1 + (1 + 1.41i)T$$
5 $$1 + (-2.12 + 0.707i)T$$
7 $$1 - T$$
good11 $$1 + 4.24T + 11T^{2}$$
13 $$1 + 6iT - 13T^{2}$$
17 $$1 - 4.24T + 17T^{2}$$
19 $$1 + 6iT - 19T^{2}$$
23 $$1 + 1.41iT - 23T^{2}$$
29 $$1 - 2.82iT - 29T^{2}$$
31 $$1 - 31T^{2}$$
37 $$1 + 6iT - 37T^{2}$$
41 $$1 + 1.41iT - 41T^{2}$$
43 $$1 - 8T + 43T^{2}$$
47 $$1 - 2.82iT - 47T^{2}$$
53 $$1 + 8.48T + 53T^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 + 10T + 61T^{2}$$
67 $$1 + 4T + 67T^{2}$$
71 $$1 - 12.7T + 71T^{2}$$
73 $$1 - 6iT - 73T^{2}$$
79 $$1 - 79T^{2}$$
83 $$1 - 2.82iT - 83T^{2}$$
89 $$1 - 7.07iT - 89T^{2}$$
97 $$1 - 6iT - 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

−10.85548826180716822731874985962, −10.27232604482245198757885544974, −9.061367907453524573939195644925, −7.982460779226936813288556093110, −7.49985951007858922781414537621, −6.25643879256575624130339364211, −5.38154317719248833473137974863, −5.02982559733495413126446465533, −2.73054577108713646790079187748, −0.78525186382678588737342378224, 1.72960187515778712577241904296, 3.13868077823481651915369999190, 4.40162955848078837620502499222, 5.33612847101818379007448476507, 6.15233619526466917963157775568, 7.81706070580839776609200741990, 9.037088269681964406689053738249, 9.846238565111668995706145436266, 10.32685703294782079381520688595, 11.16808615504926863183288843047