# Properties

 Label 2-3920-140.87-c0-0-1 Degree $2$ Conductor $3920$ Sign $0.958 + 0.286i$ Analytic cond. $1.95633$ Root an. cond. $1.39869$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.608 − 0.793i)5-s + (0.866 − 0.5i)9-s + (1.30 + 1.30i)13-s + (0.198 + 0.739i)17-s + (−0.258 − 0.965i)25-s − 1.41i·29-s + (−0.517 + 1.93i)37-s + 1.84i·41-s + (0.130 − 0.991i)45-s + (−0.366 − 1.36i)53-s + (−0.662 + 0.382i)61-s + (1.83 − 0.241i)65-s + (−0.739 + 0.198i)73-s + (0.499 − 0.866i)81-s + (0.707 + 0.292i)85-s + ⋯
 L(s)  = 1 + (0.608 − 0.793i)5-s + (0.866 − 0.5i)9-s + (1.30 + 1.30i)13-s + (0.198 + 0.739i)17-s + (−0.258 − 0.965i)25-s − 1.41i·29-s + (−0.517 + 1.93i)37-s + 1.84i·41-s + (0.130 − 0.991i)45-s + (−0.366 − 1.36i)53-s + (−0.662 + 0.382i)61-s + (1.83 − 0.241i)65-s + (−0.739 + 0.198i)73-s + (0.499 − 0.866i)81-s + (0.707 + 0.292i)85-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3920$$    =    $$2^{4} \cdot 5 \cdot 7^{2}$$ Sign: $0.958 + 0.286i$ Analytic conductor: $$1.95633$$ Root analytic conductor: $$1.39869$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3920} (3167, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3920,\ (\ :0),\ 0.958 + 0.286i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.674440431$$ $$L(\frac12)$$ $$\approx$$ $$1.674440431$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 + (-0.608 + 0.793i)T$$
7 $$1$$
good3 $$1 + (-0.866 + 0.5i)T^{2}$$
11 $$1 + (0.5 + 0.866i)T^{2}$$
13 $$1 + (-1.30 - 1.30i)T + iT^{2}$$
17 $$1 + (-0.198 - 0.739i)T + (-0.866 + 0.5i)T^{2}$$
19 $$1 + (0.5 - 0.866i)T^{2}$$
23 $$1 + (-0.866 - 0.5i)T^{2}$$
29 $$1 + 1.41iT - T^{2}$$
31 $$1 + (-0.5 - 0.866i)T^{2}$$
37 $$1 + (0.517 - 1.93i)T + (-0.866 - 0.5i)T^{2}$$
41 $$1 - 1.84iT - T^{2}$$
43 $$1 + iT^{2}$$
47 $$1 + (-0.866 - 0.5i)T^{2}$$
53 $$1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2}$$
59 $$1 + (0.5 + 0.866i)T^{2}$$
61 $$1 + (0.662 - 0.382i)T + (0.5 - 0.866i)T^{2}$$
67 $$1 + (-0.866 + 0.5i)T^{2}$$
71 $$1 - T^{2}$$
73 $$1 + (0.739 - 0.198i)T + (0.866 - 0.5i)T^{2}$$
79 $$1 + (-0.5 + 0.866i)T^{2}$$
83 $$1 - iT^{2}$$
89 $$1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2}$$
97 $$1 + (-1.30 + 1.30i)T - iT^{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

−8.470930533108821281231982259987, −8.214620851617796815037196807236, −6.95050829907999220654605834447, −6.33456887657352719276813142796, −5.84049881265685798933031272334, −4.60038746118949130947837535324, −4.25437859252723123202701610097, −3.24565989321870136426878551485, −1.80036553905066311931658170668, −1.28301217238724581554223030146, 1.23375566306066413544447781518, 2.28029721270195856769038844325, 3.24053214140901345757023793256, 3.91415239483954902311765517159, 5.17210373226385469999388904761, 5.64305920756410596978973490187, 6.50094030563071988560733070827, 7.30864893673542978583929611170, 7.71022234698632372835050622984, 8.803267991805535863481753377741