Properties

 Label 2-336-12.11-c3-0-32 Degree $2$ Conductor $336$ Sign $-0.328 + 0.944i$ Analytic cond. $19.8246$ Root an. cond. $4.45248$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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Dirichlet series

 L(s)  = 1 + (4.90 + 1.70i)3-s − 20.5i·5-s − 7i·7-s + (21.1 + 16.7i)9-s − 63.9·11-s + 56.5·13-s + (35.1 − 100. i)15-s − 81.6i·17-s − 22.4i·19-s + (11.9 − 34.3i)21-s − 114.·23-s − 298.·25-s + (75.2 + 118. i)27-s − 62.7i·29-s + 88.6i·31-s + ⋯
 L(s)  = 1 + (0.944 + 0.328i)3-s − 1.84i·5-s − 0.377i·7-s + (0.783 + 0.620i)9-s − 1.75·11-s + 1.20·13-s + (0.605 − 1.73i)15-s − 1.16i·17-s − 0.270i·19-s + (0.124 − 0.356i)21-s − 1.03·23-s − 2.38·25-s + (0.536 + 0.844i)27-s − 0.401i·29-s + 0.513i·31-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.328 + 0.944i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$336$$    =    $$2^{4} \cdot 3 \cdot 7$$ Sign: $-0.328 + 0.944i$ Analytic conductor: $$19.8246$$ Root analytic conductor: $$4.45248$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{336} (239, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 336,\ (\ :3/2),\ -0.328 + 0.944i)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$2.059018158$$ $$L(\frac12)$$ $$\approx$$ $$2.059018158$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (-4.90 - 1.70i)T$$
7 $$1 + 7iT$$
good5 $$1 + 20.5iT - 125T^{2}$$
11 $$1 + 63.9T + 1.33e3T^{2}$$
13 $$1 - 56.5T + 2.19e3T^{2}$$
17 $$1 + 81.6iT - 4.91e3T^{2}$$
19 $$1 + 22.4iT - 6.85e3T^{2}$$
23 $$1 + 114.T + 1.21e4T^{2}$$
29 $$1 + 62.7iT - 2.43e4T^{2}$$
31 $$1 - 88.6iT - 2.97e4T^{2}$$
37 $$1 + 51.3T + 5.06e4T^{2}$$
41 $$1 + 165. iT - 6.89e4T^{2}$$
43 $$1 + 393. iT - 7.95e4T^{2}$$
47 $$1 - 164.T + 1.03e5T^{2}$$
53 $$1 + 231. iT - 1.48e5T^{2}$$
59 $$1 + 13.4T + 2.05e5T^{2}$$
61 $$1 - 665.T + 2.26e5T^{2}$$
67 $$1 - 837. iT - 3.00e5T^{2}$$
71 $$1 - 175.T + 3.57e5T^{2}$$
73 $$1 - 301.T + 3.89e5T^{2}$$
79 $$1 + 1.04e3iT - 4.93e5T^{2}$$
83 $$1 - 1.26e3T + 5.71e5T^{2}$$
89 $$1 + 780. iT - 7.04e5T^{2}$$
97 $$1 + 780.T + 9.12e5T^{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

−10.60454863379908750894806815036, −9.773183195194740684333182299181, −8.788843780180822478561403241712, −8.272345023491456736252582622082, −7.41088690433566779876658261106, −5.52336438532801622451624877524, −4.77296948848676881808063822252, −3.71418400810268272415945537964, −2.15077785817859108566317828305, −0.62652500995617171403182603631, 2.03596505910109140477091251367, 2.94875801268438841338068734913, 3.84317479739909584131120145222, 5.86257241873315520567606605458, 6.62904094813203458502130489267, 7.84382849946889780766600894700, 8.203039704472934745064211451509, 9.708011552145468082388829560623, 10.51432363271099996491414579786, 11.10830587815836721068207987134