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Properties

Label 2-378-63.4-c1-0-3
Degree 22
Conductor 378378
Sign 0.609+0.792i0.609 + 0.792i
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + 0.460·5-s + (2.25 + 1.38i)7-s − 0.999·8-s + (0.230 − 0.398i)10-s + 3.64·11-s + (0.730 − 1.26i)13-s + (2.32 − 1.26i)14-s + (−0.5 + 0.866i)16-s + (1.86 − 3.23i)17-s + (−2.02 − 3.51i)19-s + (−0.230 − 0.398i)20-s + (1.82 − 3.15i)22-s − 1.13·23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + 0.205·5-s + (0.853 + 0.521i)7-s − 0.353·8-s + (0.0728 − 0.126i)10-s + 1.09·11-s + (0.202 − 0.350i)13-s + (0.621 − 0.338i)14-s + (−0.125 + 0.216i)16-s + (0.452 − 0.784i)17-s + (−0.465 − 0.805i)19-s + (−0.0514 − 0.0891i)20-s + (0.388 − 0.673i)22-s − 0.236·23-s + ⋯

Functional equation

Λ(s)=(378s/2 ΓC(s) L(s)=((0.609+0.792i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.609 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(378s/2 ΓC(s+1/2) L(s)=((0.609+0.792i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 378378    =    2⋅33⋅72 \cdot 3^{3} \cdot 7
Sign: 0.609+0.792i0.609 + 0.792i
Analytic conductor: 3.018343.01834
Root analytic conductor: 1.737331.73733
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ378(361,⋅)\chi_{378} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 378, ( :1/2), 0.609+0.792i)(2,\ 378,\ (\ :1/2),\ 0.609 + 0.792i)

Particular Values

L(1)L(1) ≈\approx 1.62452−0.800440i1.62452 - 0.800440i
L(12)L(\frac12) ≈\approx 1.62452−0.800440i1.62452 - 0.800440i
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
3 1 1
7 1+(−2.25−1.38i)T 1 + (-2.25 - 1.38i)T
good5 1−0.460T+5T2 1 - 0.460T + 5T^{2}
11 1−3.64T+11T2 1 - 3.64T + 11T^{2}
13 1+(−0.730+1.26i)T+(−6.5−11.2i)T2 1 + (-0.730 + 1.26i)T + (-6.5 - 11.2i)T^{2}
17 1+(−1.86+3.23i)T+(−8.5−14.7i)T2 1 + (-1.86 + 3.23i)T + (-8.5 - 14.7i)T^{2}
19 1+(2.02+3.51i)T+(−9.5+16.4i)T2 1 + (2.02 + 3.51i)T + (-9.5 + 16.4i)T^{2}
23 1+1.13T+23T2 1 + 1.13T + 23T^{2}
29 1+(−4.48−7.77i)T+(−14.5+25.1i)T2 1 + (-4.48 - 7.77i)T + (-14.5 + 25.1i)T^{2}
31 1+(−0.257−0.445i)T+(−15.5+26.8i)T2 1 + (-0.257 - 0.445i)T + (-15.5 + 26.8i)T^{2}
37 1+(4.55+7.88i)T+(−18.5+32.0i)T2 1 + (4.55 + 7.88i)T + (-18.5 + 32.0i)T^{2}
41 1+(−0.472+0.819i)T+(−20.5−35.5i)T2 1 + (-0.472 + 0.819i)T + (-20.5 - 35.5i)T^{2}
43 1+(−4.66−8.07i)T+(−21.5+37.2i)T2 1 + (-4.66 - 8.07i)T + (-21.5 + 37.2i)T^{2}
47 1+(−1.16+2.01i)T+(−23.5−40.7i)T2 1 + (-1.16 + 2.01i)T + (-23.5 - 40.7i)T^{2}
53 1+(6.21−10.7i)T+(−26.5−45.8i)T2 1 + (6.21 - 10.7i)T + (-26.5 - 45.8i)T^{2}
59 1+(6.44+11.1i)T+(−29.5+51.0i)T2 1 + (6.44 + 11.1i)T + (-29.5 + 51.0i)T^{2}
61 1+(6.04−10.4i)T+(−30.5−52.8i)T2 1 + (6.04 - 10.4i)T + (-30.5 - 52.8i)T^{2}
67 1+(−1.16−2.00i)T+(−33.5+58.0i)T2 1 + (-1.16 - 2.00i)T + (-33.5 + 58.0i)T^{2}
71 1+1.67T+71T2 1 + 1.67T + 71T^{2}
73 1+(6.62−11.4i)T+(−36.5−63.2i)T2 1 + (6.62 - 11.4i)T + (-36.5 - 63.2i)T^{2}
79 1+(−2.50+4.33i)T+(−39.5−68.4i)T2 1 + (-2.50 + 4.33i)T + (-39.5 - 68.4i)T^{2}
83 1+(3.32+5.75i)T+(−41.5+71.8i)T2 1 + (3.32 + 5.75i)T + (-41.5 + 71.8i)T^{2}
89 1+(−1.36−2.36i)T+(−44.5+77.0i)T2 1 + (-1.36 - 2.36i)T + (-44.5 + 77.0i)T^{2}
97 1+(5.59+9.68i)T+(−48.5+84.0i)T2 1 + (5.59 + 9.68i)T + (-48.5 + 84.0i)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

−11.32686969203929651868753031976, −10.54093751102804657654714834640, −9.349615211432034329660144714393, −8.755671667260364153794665974574, −7.48967446443218884831093625773, −6.22491978968691802364842332566, −5.22398128177574661142213873457, −4.21794801333404956715514154573, −2.83295473732128217190212672836, −1.44616186125906097032326609181, 1.69182270299532967628053410314, 3.74481133506278839689758415473, 4.50045599671527860788932802720, 5.85594484002507451364554872461, 6.60231345405632272593502687381, 7.81403251583164097095617833192, 8.444779936390306220077431060094, 9.623116016839618038712585132029, 10.57048824813704249887219649070, 11.71298199121324714001234439539

Graph of the ZZ-function along the critical line