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Properties

Label 2-378-189.20-c1-0-15
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(209,⋅)\chi_{378} (209, \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

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

Graph of the ZZ-function along the critical line