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Properties

Label 2-378-63.4-c1-0-1
Degree 22
Conductor 378378
Sign −0.609−0.792i-0.609 - 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 − 0.593·5-s + (−0.0665 + 2.64i)7-s + 0.999·8-s + (0.296 − 0.514i)10-s + 0.593·11-s + (−1.25 + 2.17i)13-s + (−2.25 − 1.38i)14-s + (−0.5 + 0.866i)16-s + (−1.46 + 2.52i)17-s + (2.69 + 4.66i)19-s + (0.296 + 0.514i)20-s + (−0.296 + 0.514i)22-s − 4.46·23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s − 0.265·5-s + (−0.0251 + 0.999i)7-s + 0.353·8-s + (0.0938 − 0.162i)10-s + 0.178·11-s + (−0.348 + 0.603i)13-s + (−0.603 − 0.368i)14-s + (−0.125 + 0.216i)16-s + (−0.354 + 0.613i)17-s + (0.617 + 1.06i)19-s + (0.0663 + 0.114i)20-s + (−0.0632 + 0.109i)22-s − 0.930·23-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: −0.609−0.792i-0.609 - 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 0.376658+0.764442i0.376658 + 0.764442i
L(12)L(\frac12) ≈\approx 0.376658+0.764442i0.376658 + 0.764442i
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+(0.0665−2.64i)T 1 + (0.0665 - 2.64i)T
good5 1+0.593T+5T2 1 + 0.593T + 5T^{2}
11 1−0.593T+11T2 1 - 0.593T + 11T^{2}
13 1+(1.25−2.17i)T+(−6.5−11.2i)T2 1 + (1.25 - 2.17i)T + (-6.5 - 11.2i)T^{2}
17 1+(1.46−2.52i)T+(−8.5−14.7i)T2 1 + (1.46 - 2.52i)T + (-8.5 - 14.7i)T^{2}
19 1+(−2.69−4.66i)T+(−9.5+16.4i)T2 1 + (-2.69 - 4.66i)T + (-9.5 + 16.4i)T^{2}
23 1+4.46T+23T2 1 + 4.46T + 23T^{2}
29 1+(−3.09−5.36i)T+(−14.5+25.1i)T2 1 + (-3.09 - 5.36i)T + (-14.5 + 25.1i)T^{2}
31 1+(−3.93−6.81i)T+(−15.5+26.8i)T2 1 + (-3.93 - 6.81i)T + (-15.5 + 26.8i)T^{2}
37 1+(−0.5−0.866i)T+(−18.5+32.0i)T2 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2}
41 1+(−0.136+0.236i)T+(−20.5−35.5i)T2 1 + (-0.136 + 0.236i)T + (-20.5 - 35.5i)T^{2}
43 1+(5.58+9.66i)T+(−21.5+37.2i)T2 1 + (5.58 + 9.66i)T + (-21.5 + 37.2i)T^{2}
47 1+(−6.08+10.5i)T+(−23.5−40.7i)T2 1 + (-6.08 + 10.5i)T + (-23.5 - 40.7i)T^{2}
53 1+(4.02−6.97i)T+(−26.5−45.8i)T2 1 + (4.02 - 6.97i)T + (-26.5 - 45.8i)T^{2}
59 1+(−4.32−7.48i)T+(−29.5+51.0i)T2 1 + (-4.32 - 7.48i)T + (-29.5 + 51.0i)T^{2}
61 1+(−3.32+5.75i)T+(−30.5−52.8i)T2 1 + (-3.32 + 5.75i)T + (-30.5 - 52.8i)T^{2}
67 1+(−0.956−1.65i)T+(−33.5+58.0i)T2 1 + (-0.956 - 1.65i)T + (-33.5 + 58.0i)T^{2}
71 1−14.4T+71T2 1 - 14.4T + 71T^{2}
73 1+(−3.95+6.85i)T+(−36.5−63.2i)T2 1 + (-3.95 + 6.85i)T + (-36.5 - 63.2i)T^{2}
79 1+(−4.62+8.00i)T+(−39.5−68.4i)T2 1 + (-4.62 + 8.00i)T + (-39.5 - 68.4i)T^{2}
83 1+(3.85+6.66i)T+(−41.5+71.8i)T2 1 + (3.85 + 6.66i)T + (-41.5 + 71.8i)T^{2}
89 1+(−6.21−10.7i)T+(−44.5+77.0i)T2 1 + (-6.21 - 10.7i)T + (-44.5 + 77.0i)T^{2}
97 1+(−5.86−10.1i)T+(−48.5+84.0i)T2 1 + (-5.86 - 10.1i)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

−11.98647974666041346932303693560, −10.57594986926823079797590206614, −9.727104775925431944125837671449, −8.755022131637625542872844524264, −8.113658215841622770996367541887, −6.95522896531781704814822976055, −6.02304779982039994760018645355, −5.06043877812093635130566172818, −3.69721634376078247122664610306, −1.92734219114311631730865400329, 0.64848224129327926610869568510, 2.55147765920002646137517570120, 3.86472451637733685311821579134, 4.81664585564264533858711100717, 6.39360013178579474315822769517, 7.56037032028413341592125498858, 8.135812797060768007880511251695, 9.597382888290522271631012122468, 9.956711435928012677571310782686, 11.21847873660925762458660780044

Graph of the ZZ-function along the critical line