# Properties

 Label 2-768-48.29-c0-0-3 Degree $2$ Conductor $768$ Sign $-0.382 + 0.923i$ Analytic cond. $0.383281$ Root an. cond. $0.619097$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−0.707 − 0.707i)3-s − 1.41i·7-s + 1.00i·9-s + (−1 − i)13-s + (−1.00 + 1.00i)21-s − i·25-s + (0.707 − 0.707i)27-s − 1.41·31-s + (1 − i)37-s + 1.41i·39-s − 1.00·49-s + (1 + i)61-s + 1.41·63-s + (1.41 + 1.41i)67-s + (−0.707 + 0.707i)75-s + ⋯
 L(s)  = 1 + (−0.707 − 0.707i)3-s − 1.41i·7-s + 1.00i·9-s + (−1 − i)13-s + (−1.00 + 1.00i)21-s − i·25-s + (0.707 − 0.707i)27-s − 1.41·31-s + (1 − i)37-s + 1.41i·39-s − 1.00·49-s + (1 + i)61-s + 1.41·63-s + (1.41 + 1.41i)67-s + (−0.707 + 0.707i)75-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$768$$    =    $$2^{8} \cdot 3$$ Sign: $-0.382 + 0.923i$ Analytic conductor: $$0.383281$$ Root analytic conductor: $$0.619097$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{768} (65, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 768,\ (\ :0),\ -0.382 + 0.923i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.6551410767$$ $$L(\frac12)$$ $$\approx$$ $$0.6551410767$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (0.707 + 0.707i)T$$
good5 $$1 + iT^{2}$$
7 $$1 + 1.41iT - T^{2}$$
11 $$1 + iT^{2}$$
13 $$1 + (1 + i)T + iT^{2}$$
17 $$1 - T^{2}$$
19 $$1 + iT^{2}$$
23 $$1 + T^{2}$$
29 $$1 - iT^{2}$$
31 $$1 + 1.41T + T^{2}$$
37 $$1 + (-1 + i)T - iT^{2}$$
41 $$1 + T^{2}$$
43 $$1 - iT^{2}$$
47 $$1 - T^{2}$$
53 $$1 + iT^{2}$$
59 $$1 + iT^{2}$$
61 $$1 + (-1 - i)T + iT^{2}$$
67 $$1 + (-1.41 - 1.41i)T + iT^{2}$$
71 $$1 + T^{2}$$
73 $$1 - T^{2}$$
79 $$1 - 1.41T + T^{2}$$
83 $$1 - iT^{2}$$
89 $$1 + T^{2}$$
97 $$1 + 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

−10.45654648858001826076431688013, −9.674456822114586874000020282693, −8.235966008383804841800828033247, −7.47448400418578472126877030440, −6.97786142857710253982331009585, −5.86771939584615238294315056329, −4.95346695150744839011667156222, −3.90513823797067780262373371644, −2.38076276436362608287248420143, −0.75137044797525364260919094343, 2.10110209145205183716226362850, 3.43294835580117660338348996374, 4.70122051681237414997937803449, 5.37356384890399184672899423801, 6.24251051548040817548455498901, 7.17887634069304071626328509295, 8.466911060945603222420061089713, 9.441288557782877988434647510542, 9.626310323968048748989983933636, 10.97906695086689203669690864401