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Properties

Label 2-378-189.104-c1-0-13
Degree 22
Conductor 378378
Sign 0.853+0.521i0.853 + 0.521i
Analytic cond. 3.018343.01834
Root an. cond. 1.737331.73733
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.342 + 0.939i)2-s + (0.0584 − 1.73i)3-s + (−0.766 − 0.642i)4-s + (−0.439 + 2.49i)5-s + (1.60 + 0.646i)6-s + (0.411 − 2.61i)7-s + (0.866 − 0.500i)8-s + (−2.99 − 0.202i)9-s + (−2.19 − 1.26i)10-s + (5.67 − 1.00i)11-s + (−1.15 + 1.28i)12-s + (−0.939 − 2.58i)13-s + (2.31 + 1.28i)14-s + (4.28 + 0.906i)15-s + (0.173 + 0.984i)16-s + (1.32 − 2.29i)17-s + ⋯
L(s)  = 1  + (−0.241 + 0.664i)2-s + (0.0337 − 0.999i)3-s + (−0.383 − 0.321i)4-s + (−0.196 + 1.11i)5-s + (0.655 + 0.264i)6-s + (0.155 − 0.987i)7-s + (0.306 − 0.176i)8-s + (−0.997 − 0.0674i)9-s + (−0.693 − 0.400i)10-s + (1.71 − 0.301i)11-s + (−0.334 + 0.371i)12-s + (−0.260 − 0.715i)13-s + (0.618 + 0.342i)14-s + (1.10 + 0.234i)15-s + (0.0434 + 0.246i)16-s + (0.320 − 0.555i)17-s + ⋯

Functional equation

Λ(s)=(378s/2 ΓC(s) L(s)=((0.853+0.521i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.521i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(378s/2 ΓC(s+1/2) L(s)=((0.853+0.521i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.853 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 378378    =    2⋅33⋅72 \cdot 3^{3} \cdot 7
Sign: 0.853+0.521i0.853 + 0.521i
Analytic conductor: 3.018343.01834
Root analytic conductor: 1.737331.73733
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ378(293,⋅)\chi_{378} (293, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 378, ( :1/2), 0.853+0.521i)(2,\ 378,\ (\ :1/2),\ 0.853 + 0.521i)

Particular Values

L(1)L(1) ≈\approx 1.11381−0.313270i1.11381 - 0.313270i
L(12)L(\frac12) ≈\approx 1.11381−0.313270i1.11381 - 0.313270i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.342−0.939i)T 1 + (0.342 - 0.939i)T
3 1+(−0.0584+1.73i)T 1 + (-0.0584 + 1.73i)T
7 1+(−0.411+2.61i)T 1 + (-0.411 + 2.61i)T
good5 1+(0.439−2.49i)T+(−4.69−1.71i)T2 1 + (0.439 - 2.49i)T + (-4.69 - 1.71i)T^{2}
11 1+(−5.67+1.00i)T+(10.3−3.76i)T2 1 + (-5.67 + 1.00i)T + (10.3 - 3.76i)T^{2}
13 1+(0.939+2.58i)T+(−9.95+8.35i)T2 1 + (0.939 + 2.58i)T + (-9.95 + 8.35i)T^{2}
17 1+(−1.32+2.29i)T+(−8.5−14.7i)T2 1 + (-1.32 + 2.29i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.0684−0.0395i)T+(9.5−16.4i)T2 1 + (0.0684 - 0.0395i)T + (9.5 - 16.4i)T^{2}
23 1+(−0.0439+0.0524i)T+(−3.99−22.6i)T2 1 + (-0.0439 + 0.0524i)T + (-3.99 - 22.6i)T^{2}
29 1+(−3.25+8.94i)T+(−22.2−18.6i)T2 1 + (-3.25 + 8.94i)T + (-22.2 - 18.6i)T^{2}
31 1+(−6.33+7.55i)T+(−5.38−30.5i)T2 1 + (-6.33 + 7.55i)T + (-5.38 - 30.5i)T^{2}
37 1+(2.36−4.08i)T+(−18.5−32.0i)T2 1 + (2.36 - 4.08i)T + (-18.5 - 32.0i)T^{2}
41 1+(−2.92+1.06i)T+(31.4−26.3i)T2 1 + (-2.92 + 1.06i)T + (31.4 - 26.3i)T^{2}
43 1+(−0.166−0.944i)T+(−40.4+14.7i)T2 1 + (-0.166 - 0.944i)T + (-40.4 + 14.7i)T^{2}
47 1+(5.94−4.98i)T+(8.16−46.2i)T2 1 + (5.94 - 4.98i)T + (8.16 - 46.2i)T^{2}
53 1−11.8iT−53T2 1 - 11.8iT - 53T^{2}
59 1+(1.48−8.42i)T+(−55.4−20.1i)T2 1 + (1.48 - 8.42i)T + (-55.4 - 20.1i)T^{2}
61 1+(1.31+1.57i)T+(−10.5+60.0i)T2 1 + (1.31 + 1.57i)T + (-10.5 + 60.0i)T^{2}
67 1+(0.412−0.150i)T+(51.3−43.0i)T2 1 + (0.412 - 0.150i)T + (51.3 - 43.0i)T^{2}
71 1+(6.39+3.69i)T+(35.5+61.4i)T2 1 + (6.39 + 3.69i)T + (35.5 + 61.4i)T^{2}
73 1+(7.57−4.37i)T+(36.5−63.2i)T2 1 + (7.57 - 4.37i)T + (36.5 - 63.2i)T^{2}
79 1+(1.93+0.705i)T+(60.5+50.7i)T2 1 + (1.93 + 0.705i)T + (60.5 + 50.7i)T^{2}
83 1+(−4.09−1.49i)T+(63.5+53.3i)T2 1 + (-4.09 - 1.49i)T + (63.5 + 53.3i)T^{2}
89 1+(−8.31−14.4i)T+(−44.5+77.0i)T2 1 + (-8.31 - 14.4i)T + (-44.5 + 77.0i)T^{2}
97 1+(5.68−1.00i)T+(91.1−33.1i)T2 1 + (5.68 - 1.00i)T + (91.1 - 33.1i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−11.37815332544280480957583097081, −10.35721012397476659212783942784, −9.364728106490847286470313899677, −8.069875050607039527036532093167, −7.47219702379095233051936466129, −6.62310462775174033904720884785, −6.05504232408160498740438388215, −4.27030702373740102941158053143, −2.93810148645942692730068145126, −0.968472694167237133551123087728, 1.61170324443451594516637282995, 3.36046830105826253222213362033, 4.46454359907457573321090403826, 5.16142369760757327318067494574, 6.56994854635155305544233061274, 8.420130359087124581151691278985, 8.871718741222762696345535118052, 9.438606639767289394263081903738, 10.44517567636474008645574404691, 11.69345039281596206440778613324

Graph of the ZZ-function along the critical line