Properties

 Label 2-378-21.17-c1-0-9 Degree $2$ Conductor $378$ Sign $-0.507 + 0.861i$ Analytic cond. $3.01834$ Root an. cond. $1.73733$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−1.22 − 2.12i)5-s + (−1 − 2.44i)7-s − 0.999i·8-s + (−2.12 − 1.22i)10-s + (−3.67 − 2.12i)11-s + 4.18i·13-s + (−2.09 − 1.62i)14-s + (−0.5 − 0.866i)16-s + (−1.22 + 2.12i)17-s + (4.24 − 2.44i)19-s − 2.44·20-s − 4.24·22-s + (5.19 − 3i)23-s + ⋯
 L(s)  = 1 + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.547 − 0.948i)5-s + (−0.377 − 0.925i)7-s − 0.353i·8-s + (−0.670 − 0.387i)10-s + (−1.10 − 0.639i)11-s + 1.15i·13-s + (−0.558 − 0.433i)14-s + (−0.125 − 0.216i)16-s + (−0.297 + 0.514i)17-s + (0.973 − 0.561i)19-s − 0.547·20-s − 0.904·22-s + (1.08 − 0.625i)23-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$378$$    =    $$2 \cdot 3^{3} \cdot 7$$ Sign: $-0.507 + 0.861i$ Analytic conductor: $$3.01834$$ Root analytic conductor: $$1.73733$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{378} (269, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 378,\ (\ :1/2),\ -0.507 + 0.861i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.725775 - 1.27061i$$ $$L(\frac12)$$ $$\approx$$ $$0.725775 - 1.27061i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.866 + 0.5i)T$$
3 $$1$$
7 $$1 + (1 + 2.44i)T$$
good5 $$1 + (1.22 + 2.12i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (3.67 + 2.12i)T + (5.5 + 9.52i)T^{2}$$
13 $$1 - 4.18iT - 13T^{2}$$
17 $$1 + (1.22 - 2.12i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-4.24 + 2.44i)T + (9.5 - 16.4i)T^{2}$$
23 $$1 + (-5.19 + 3i)T + (11.5 - 19.9i)T^{2}$$
29 $$1 + 10.2iT - 29T^{2}$$
31 $$1 + (-4.86 - 2.80i)T + (15.5 + 26.8i)T^{2}$$
37 $$1 + (-1.62 - 2.80i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + 2.44T + 41T^{2}$$
43 $$1 - 7T + 43T^{2}$$
47 $$1 + (-3.97 - 6.87i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-2.15 - 1.24i)T + (26.5 + 45.8i)T^{2}$$
59 $$1 + (-1.22 + 2.12i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (0.621 - 0.358i)T + (30.5 - 52.8i)T^{2}$$
67 $$1 + (-1.74 + 3.01i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 - 12.7iT - 71T^{2}$$
73 $$1 + (13.2 + 7.64i)T + (36.5 + 63.2i)T^{2}$$
79 $$1 + (4.62 + 8.00i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + 5.49T + 83T^{2}$$
89 $$1 + (-8.87 - 15.3i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + 6.63iT - 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

−11.20081687807881202731732554808, −10.33614874097371869673844274233, −9.287233216963825749759262131074, −8.266292611141300835988971129589, −7.24677746307212328023735942723, −6.12783474940424461333674694372, −4.79968519774219734071919844863, −4.18672145210840021439404454344, −2.81855434303915579719796488249, −0.824009368639995085470008281421, 2.70371834603069729774990166611, 3.33064832592204535138309223376, 5.04028860507216640092781304137, 5.73122122292843778804414798441, 7.10180871557025749508183412722, 7.57190389867283249525724145430, 8.757699378618956612764061741524, 9.997297079003790759395636516219, 10.86954330708787085577898820478, 11.77091299779775380806505818143