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Properties

Label 2-378-189.104-c1-0-12
Degree 22
Conductor 378378
Sign 0.907−0.419i0.907 - 0.419i
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 + (1.51 − 0.847i)3-s + (−0.766 − 0.642i)4-s + (−0.0403 + 0.228i)5-s + (0.279 + 1.70i)6-s + (1.82 + 1.91i)7-s + (0.866 − 0.500i)8-s + (1.56 − 2.55i)9-s + (−0.201 − 0.116i)10-s + (2.79 − 0.492i)11-s + (−1.70 − 0.322i)12-s + (−1.29 − 3.56i)13-s + (−2.42 + 1.06i)14-s + (0.132 + 0.379i)15-s + (0.173 + 0.984i)16-s + (−3.85 + 6.67i)17-s + ⋯
L(s)  = 1  + (−0.241 + 0.664i)2-s + (0.872 − 0.489i)3-s + (−0.383 − 0.321i)4-s + (−0.0180 + 0.102i)5-s + (0.114 + 0.697i)6-s + (0.690 + 0.723i)7-s + (0.306 − 0.176i)8-s + (0.521 − 0.853i)9-s + (−0.0636 − 0.0367i)10-s + (0.841 − 0.148i)11-s + (−0.491 − 0.0929i)12-s + (−0.359 − 0.987i)13-s + (−0.647 + 0.283i)14-s + (0.0342 + 0.0980i)15-s + (0.0434 + 0.246i)16-s + (−0.935 + 1.62i)17-s + ⋯

Functional equation

Λ(s)=(378s/2 ΓC(s) L(s)=((0.907−0.419i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(378s/2 ΓC(s+1/2) L(s)=((0.907−0.419i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 378378    =    2⋅33⋅72 \cdot 3^{3} \cdot 7
Sign: 0.907−0.419i0.907 - 0.419i
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), 0.907−0.419i)(2,\ 378,\ (\ :1/2),\ 0.907 - 0.419i)

Particular Values

L(1)L(1) ≈\approx 1.63280+0.359494i1.63280 + 0.359494i
L(12)L(\frac12) ≈\approx 1.63280+0.359494i1.63280 + 0.359494i
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+(−1.51+0.847i)T 1 + (-1.51 + 0.847i)T
7 1+(−1.82−1.91i)T 1 + (-1.82 - 1.91i)T
good5 1+(0.0403−0.228i)T+(−4.69−1.71i)T2 1 + (0.0403 - 0.228i)T + (-4.69 - 1.71i)T^{2}
11 1+(−2.79+0.492i)T+(10.3−3.76i)T2 1 + (-2.79 + 0.492i)T + (10.3 - 3.76i)T^{2}
13 1+(1.29+3.56i)T+(−9.95+8.35i)T2 1 + (1.29 + 3.56i)T + (-9.95 + 8.35i)T^{2}
17 1+(3.85−6.67i)T+(−8.5−14.7i)T2 1 + (3.85 - 6.67i)T + (-8.5 - 14.7i)T^{2}
19 1+(−2.59+1.49i)T+(9.5−16.4i)T2 1 + (-2.59 + 1.49i)T + (9.5 - 16.4i)T^{2}
23 1+(−2.84+3.38i)T+(−3.99−22.6i)T2 1 + (-2.84 + 3.38i)T + (-3.99 - 22.6i)T^{2}
29 1+(2.66−7.31i)T+(−22.2−18.6i)T2 1 + (2.66 - 7.31i)T + (-22.2 - 18.6i)T^{2}
31 1+(−0.0308+0.0367i)T+(−5.38−30.5i)T2 1 + (-0.0308 + 0.0367i)T + (-5.38 - 30.5i)T^{2}
37 1+(−5.34+9.25i)T+(−18.5−32.0i)T2 1 + (-5.34 + 9.25i)T + (-18.5 - 32.0i)T^{2}
41 1+(5.22−1.90i)T+(31.4−26.3i)T2 1 + (5.22 - 1.90i)T + (31.4 - 26.3i)T^{2}
43 1+(0.0109+0.0621i)T+(−40.4+14.7i)T2 1 + (0.0109 + 0.0621i)T + (-40.4 + 14.7i)T^{2}
47 1+(7.18−6.02i)T+(8.16−46.2i)T2 1 + (7.18 - 6.02i)T + (8.16 - 46.2i)T^{2}
53 1+10.8iT−53T2 1 + 10.8iT - 53T^{2}
59 1+(1.35−7.65i)T+(−55.4−20.1i)T2 1 + (1.35 - 7.65i)T + (-55.4 - 20.1i)T^{2}
61 1+(5.20+6.20i)T+(−10.5+60.0i)T2 1 + (5.20 + 6.20i)T + (-10.5 + 60.0i)T^{2}
67 1+(13.8−5.02i)T+(51.3−43.0i)T2 1 + (13.8 - 5.02i)T + (51.3 - 43.0i)T^{2}
71 1+(6.22+3.59i)T+(35.5+61.4i)T2 1 + (6.22 + 3.59i)T + (35.5 + 61.4i)T^{2}
73 1+(7.95−4.59i)T+(36.5−63.2i)T2 1 + (7.95 - 4.59i)T + (36.5 - 63.2i)T^{2}
79 1+(9.36+3.40i)T+(60.5+50.7i)T2 1 + (9.36 + 3.40i)T + (60.5 + 50.7i)T^{2}
83 1+(3.51+1.27i)T+(63.5+53.3i)T2 1 + (3.51 + 1.27i)T + (63.5 + 53.3i)T^{2}
89 1+(2.75+4.76i)T+(−44.5+77.0i)T2 1 + (2.75 + 4.76i)T + (-44.5 + 77.0i)T^{2}
97 1+(−9.53+1.68i)T+(91.1−33.1i)T2 1 + (-9.53 + 1.68i)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

−11.39095689700699069316433198220, −10.38914834557517532651083709911, −9.022262047086704063361489311715, −8.737645059556231258544261497641, −7.76328543865458478377088616106, −6.84908494646690683765745599392, −5.85592847762170377640153090112, −4.55778986536108085852092749261, −3.10276681451199369800203648636, −1.56588819047187209540073691440, 1.56980422985595043121037958902, 2.95277981230077951627846455471, 4.27459841924713271031577165033, 4.80963515851761506077918012964, 6.92566031262644403748282213137, 7.70751843322327597583969566880, 8.859330333697841430929646768333, 9.437515691177376624835931880539, 10.23230145400498204446216429226, 11.43571384867623627945615138966

Graph of the ZZ-function along the critical line