Properties

 Label 2-1368-152.123-c0-0-0 Degree $2$ Conductor $1368$ Sign $0.944 + 0.327i$ Analytic cond. $0.682720$ Root an. cond. $0.826269$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.5 + 0.866i)8-s + (0.766 + 1.32i)11-s + (0.766 − 0.642i)16-s + (0.173 + 0.984i)17-s + (−0.939 + 0.342i)19-s + (1.17 − 0.984i)22-s + (0.766 + 0.642i)25-s + (−0.766 − 0.642i)32-s + (0.939 − 0.342i)34-s + (0.5 + 0.866i)38-s + (1.43 − 1.20i)41-s + (0.939 + 0.342i)43-s + (−1.17 − 0.984i)44-s + ⋯
 L(s)  = 1 + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.5 + 0.866i)8-s + (0.766 + 1.32i)11-s + (0.766 − 0.642i)16-s + (0.173 + 0.984i)17-s + (−0.939 + 0.342i)19-s + (1.17 − 0.984i)22-s + (0.766 + 0.642i)25-s + (−0.766 − 0.642i)32-s + (0.939 − 0.342i)34-s + (0.5 + 0.866i)38-s + (1.43 − 1.20i)41-s + (0.939 + 0.342i)43-s + (−1.17 − 0.984i)44-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$1368$$    =    $$2^{3} \cdot 3^{2} \cdot 19$$ Sign: $0.944 + 0.327i$ Analytic conductor: $$0.682720$$ Root analytic conductor: $$0.826269$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1368} (883, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1368,\ (\ :0),\ 0.944 + 0.327i)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.9100432835$$ $$L(\frac12)$$ $$\approx$$ $$0.9100432835$$ $$L(1)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.173 + 0.984i)T$$
3 $$1$$
19 $$1 + (0.939 - 0.342i)T$$
good5 $$1 + (-0.766 - 0.642i)T^{2}$$
7 $$1 + (0.5 + 0.866i)T^{2}$$
11 $$1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2}$$
13 $$1 + (-0.173 - 0.984i)T^{2}$$
17 $$1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2}$$
23 $$1 + (-0.766 + 0.642i)T^{2}$$
29 $$1 + (0.939 + 0.342i)T^{2}$$
31 $$1 + (0.5 + 0.866i)T^{2}$$
37 $$1 - T^{2}$$
41 $$1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2}$$
43 $$1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2}$$
47 $$1 + (0.939 + 0.342i)T^{2}$$
53 $$1 + (-0.766 + 0.642i)T^{2}$$
59 $$1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2}$$
61 $$1 + (-0.766 + 0.642i)T^{2}$$
67 $$1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2}$$
71 $$1 + (-0.766 - 0.642i)T^{2}$$
73 $$1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2}$$
79 $$1 + (-0.173 + 0.984i)T^{2}$$
83 $$1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2}$$
89 $$1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2}$$
97 $$1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2}$$
show less
$$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

−9.771364813906417495643272131362, −9.144789977060353409155702329344, −8.351039223547349694130518410890, −7.46493039548645712787797255844, −6.51197585659926707961430408301, −5.36270795373674529650893321116, −4.32776739984497326455100462331, −3.75770325074225807551660191206, −2.39820065182450849368983229293, −1.47620533384632479922595983506, 0.936260874972603277636415509724, 2.86846714021961662671051209902, 4.06014770763908497093717848947, 4.87527718899260544314940461264, 5.97434646435689401033691171678, 6.43916642903390883214737042578, 7.39287233844047163327224317642, 8.217506813752072512782278837207, 8.972802786353350637828390561469, 9.442599146872639339605450384510