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Properties

Label 2-378-27.4-c1-0-3
Degree 22
Conductor 378378
Sign 0.0984−0.995i0.0984 - 0.995i
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  + (0.939 + 0.342i)2-s + (−1.33 − 1.10i)3-s + (0.766 + 0.642i)4-s + (−0.412 + 2.33i)5-s + (−0.874 − 1.49i)6-s + (−0.766 + 0.642i)7-s + (0.500 + 0.866i)8-s + (0.555 + 2.94i)9-s + (−1.18 + 2.05i)10-s + (0.330 + 1.87i)11-s + (−0.310 − 1.70i)12-s + (−0.325 + 0.118i)13-s + (−0.939 + 0.342i)14-s + (3.13 − 2.66i)15-s + (0.173 + 0.984i)16-s + (−1.98 + 3.43i)17-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (−0.769 − 0.638i)3-s + (0.383 + 0.321i)4-s + (−0.184 + 1.04i)5-s + (−0.357 − 0.610i)6-s + (−0.289 + 0.242i)7-s + (0.176 + 0.306i)8-s + (0.185 + 0.982i)9-s + (−0.375 + 0.650i)10-s + (0.0997 + 0.565i)11-s + (−0.0897 − 0.491i)12-s + (−0.0902 + 0.0328i)13-s + (−0.251 + 0.0914i)14-s + (0.809 − 0.687i)15-s + (0.0434 + 0.246i)16-s + (−0.480 + 0.832i)17-s + ⋯

Functional equation

Λ(s)=(378s/2 ΓC(s) L(s)=((0.0984−0.995i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0984 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(378s/2 ΓC(s+1/2) L(s)=((0.0984−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0984 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 378378    =    2⋅33⋅72 \cdot 3^{3} \cdot 7
Sign: 0.0984−0.995i0.0984 - 0.995i
Analytic conductor: 3.018343.01834
Root analytic conductor: 1.737331.73733
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ378(85,⋅)\chi_{378} (85, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 378, ( :1/2), 0.0984−0.995i)(2,\ 378,\ (\ :1/2),\ 0.0984 - 0.995i)

Particular Values

L(1)L(1) ≈\approx 0.986152+0.893376i0.986152 + 0.893376i
L(12)L(\frac12) ≈\approx 0.986152+0.893376i0.986152 + 0.893376i
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+(−0.939−0.342i)T 1 + (-0.939 - 0.342i)T
3 1+(1.33+1.10i)T 1 + (1.33 + 1.10i)T
7 1+(0.766−0.642i)T 1 + (0.766 - 0.642i)T
good5 1+(0.412−2.33i)T+(−4.69−1.71i)T2 1 + (0.412 - 2.33i)T + (-4.69 - 1.71i)T^{2}
11 1+(−0.330−1.87i)T+(−10.3+3.76i)T2 1 + (-0.330 - 1.87i)T + (-10.3 + 3.76i)T^{2}
13 1+(0.325−0.118i)T+(9.95−8.35i)T2 1 + (0.325 - 0.118i)T + (9.95 - 8.35i)T^{2}
17 1+(1.98−3.43i)T+(−8.5−14.7i)T2 1 + (1.98 - 3.43i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.954−1.65i)T+(−9.5+16.4i)T2 1 + (-0.954 - 1.65i)T + (-9.5 + 16.4i)T^{2}
23 1+(−1.91−1.60i)T+(3.99+22.6i)T2 1 + (-1.91 - 1.60i)T + (3.99 + 22.6i)T^{2}
29 1+(−7.53−2.74i)T+(22.2+18.6i)T2 1 + (-7.53 - 2.74i)T + (22.2 + 18.6i)T^{2}
31 1+(3.82+3.20i)T+(5.38+30.5i)T2 1 + (3.82 + 3.20i)T + (5.38 + 30.5i)T^{2}
37 1+(0.898−1.55i)T+(−18.5−32.0i)T2 1 + (0.898 - 1.55i)T + (-18.5 - 32.0i)T^{2}
41 1+(8.54−3.10i)T+(31.4−26.3i)T2 1 + (8.54 - 3.10i)T + (31.4 - 26.3i)T^{2}
43 1+(1.74+9.89i)T+(−40.4+14.7i)T2 1 + (1.74 + 9.89i)T + (-40.4 + 14.7i)T^{2}
47 1+(−9.88+8.29i)T+(8.16−46.2i)T2 1 + (-9.88 + 8.29i)T + (8.16 - 46.2i)T^{2}
53 1+2.36T+53T2 1 + 2.36T + 53T^{2}
59 1+(−0.541+3.07i)T+(−55.4−20.1i)T2 1 + (-0.541 + 3.07i)T + (-55.4 - 20.1i)T^{2}
61 1+(2.82−2.37i)T+(10.5−60.0i)T2 1 + (2.82 - 2.37i)T + (10.5 - 60.0i)T^{2}
67 1+(−3.72+1.35i)T+(51.3−43.0i)T2 1 + (-3.72 + 1.35i)T + (51.3 - 43.0i)T^{2}
71 1+(0.671−1.16i)T+(−35.5−61.4i)T2 1 + (0.671 - 1.16i)T + (-35.5 - 61.4i)T^{2}
73 1+(4.74+8.22i)T+(−36.5+63.2i)T2 1 + (4.74 + 8.22i)T + (-36.5 + 63.2i)T^{2}
79 1+(3.98+1.45i)T+(60.5+50.7i)T2 1 + (3.98 + 1.45i)T + (60.5 + 50.7i)T^{2}
83 1+(−7.30−2.65i)T+(63.5+53.3i)T2 1 + (-7.30 - 2.65i)T + (63.5 + 53.3i)T^{2}
89 1+(−7.20−12.4i)T+(−44.5+77.0i)T2 1 + (-7.20 - 12.4i)T + (-44.5 + 77.0i)T^{2}
97 1+(2.51+14.2i)T+(−91.1+33.1i)T2 1 + (2.51 + 14.2i)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

−11.78220286460410847518183842954, −10.80973933152597441307376723942, −10.17121634943076061247319124208, −8.554595012345043829907292121806, −7.33006896303218411804832917890, −6.79803464419227001437035941825, −5.95864086854930525236925495983, −4.86239668729538213259686164056, −3.46705239413104855360979427644, −2.08474523955620973701236294486, 0.814441443001673490886205861847, 3.12415147646527095042214825404, 4.43286777002157351550230606138, 4.98739260616200007541400580960, 6.07004791011245640196753896488, 7.07085013340698062600879695189, 8.625640619835064558046279804789, 9.428090092074027714360439402362, 10.44632719835734938454900205995, 11.27813675563432963432693326323

Graph of the ZZ-function along the critical line