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Properties

Label 2-378-63.4-c1-0-5
Degree 22
Conductor 378378
Sign 0.999+0.00294i0.999 + 0.00294i
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 + 3.18·5-s + (0.710 − 2.54i)7-s + 0.999·8-s + (−1.59 + 2.75i)10-s − 3.18·11-s + (2.85 − 4.93i)13-s + (1.85 + 1.88i)14-s + (−0.5 + 0.866i)16-s + (0.760 − 1.31i)17-s + (−0.641 − 1.11i)19-s + (−1.59 − 2.75i)20-s + (1.59 − 2.75i)22-s − 2.23·23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + 1.42·5-s + (0.268 − 0.963i)7-s + 0.353·8-s + (−0.503 + 0.871i)10-s − 0.959·11-s + (0.790 − 1.36i)13-s + (0.494 + 0.505i)14-s + (−0.125 + 0.216i)16-s + (0.184 − 0.319i)17-s + (−0.147 − 0.254i)19-s + (−0.355 − 0.616i)20-s + (0.339 − 0.587i)22-s − 0.466·23-s + ⋯

Functional equation

Λ(s)=(378s/2 ΓC(s) L(s)=((0.999+0.00294i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00294i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(378s/2 ΓC(s+1/2) L(s)=((0.999+0.00294i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00294i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 378378    =    2⋅33⋅72 \cdot 3^{3} \cdot 7
Sign: 0.999+0.00294i0.999 + 0.00294i
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), 0.999+0.00294i)(2,\ 378,\ (\ :1/2),\ 0.999 + 0.00294i)

Particular Values

L(1)L(1) ≈\approx 1.38973−0.00204684i1.38973 - 0.00204684i
L(12)L(\frac12) ≈\approx 1.38973−0.00204684i1.38973 - 0.00204684i
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+(−0.710+2.54i)T 1 + (-0.710 + 2.54i)T
good5 1−3.18T+5T2 1 - 3.18T + 5T^{2}
11 1+3.18T+11T2 1 + 3.18T + 11T^{2}
13 1+(−2.85+4.93i)T+(−6.5−11.2i)T2 1 + (-2.85 + 4.93i)T + (-6.5 - 11.2i)T^{2}
17 1+(−0.760+1.31i)T+(−8.5−14.7i)T2 1 + (-0.760 + 1.31i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.641+1.11i)T+(−9.5+16.4i)T2 1 + (0.641 + 1.11i)T + (-9.5 + 16.4i)T^{2}
23 1+2.23T+23T2 1 + 2.23T + 23T^{2}
29 1+(−3.54−6.13i)T+(−14.5+25.1i)T2 1 + (-3.54 - 6.13i)T + (-14.5 + 25.1i)T^{2}
31 1+(−4.71−8.15i)T+(−15.5+26.8i)T2 1 + (-4.71 - 8.15i)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+(−2.80+4.85i)T+(−20.5−35.5i)T2 1 + (-2.80 + 4.85i)T + (-20.5 - 35.5i)T^{2}
43 1+(−3.41−5.91i)T+(−21.5+37.2i)T2 1 + (-3.41 - 5.91i)T + (-21.5 + 37.2i)T^{2}
47 1+(2.91−5.04i)T+(−23.5−40.7i)T2 1 + (2.91 - 5.04i)T + (-23.5 - 40.7i)T^{2}
53 1+(1.02−1.78i)T+(−26.5−45.8i)T2 1 + (1.02 - 1.78i)T + (-26.5 - 45.8i)T^{2}
59 1+(0.562+0.974i)T+(−29.5+51.0i)T2 1 + (0.562 + 0.974i)T + (-29.5 + 51.0i)T^{2}
61 1+(1.56−2.70i)T+(−30.5−52.8i)T2 1 + (1.56 - 2.70i)T + (-30.5 - 52.8i)T^{2}
67 1+(5.48+9.49i)T+(−33.5+58.0i)T2 1 + (5.48 + 9.49i)T + (-33.5 + 58.0i)T^{2}
71 1+8.69T+71T2 1 + 8.69T + 71T^{2}
73 1+(2.48−4.30i)T+(−36.5−63.2i)T2 1 + (2.48 - 4.30i)T + (-36.5 - 63.2i)T^{2}
79 1+(−2.06+3.58i)T+(−39.5−68.4i)T2 1 + (-2.06 + 3.58i)T + (-39.5 - 68.4i)T^{2}
83 1+(−4.03−6.98i)T+(−41.5+71.8i)T2 1 + (-4.03 - 6.98i)T + (-41.5 + 71.8i)T^{2}
89 1+(0.112+0.195i)T+(−44.5+77.0i)T2 1 + (0.112 + 0.195i)T + (-44.5 + 77.0i)T^{2}
97 1+(−7.42−12.8i)T+(−48.5+84.0i)T2 1 + (-7.42 - 12.8i)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

−10.65854808284521445423340039810, −10.56567793432145524084938723653, −9.603360608405160181028994887321, −8.485915307753444226217294737256, −7.66682594770704328895198652806, −6.56306645923745034013988641708, −5.65171965761350440612036963254, −4.81146958152562143543122396892, −2.97002455780179579234501584556, −1.21296200663693563952143453751, 1.80486715827586510244283793887, 2.58332120179096718550293936343, 4.35302998245710172781996760259, 5.67587743052420171466070092200, 6.33494758858488591759251296799, 7.959176723345529284096229554916, 8.817130951228127687827628812753, 9.634404470272099906715239586508, 10.28982942262321705074592652744, 11.35899864675022016392162119692

Graph of the ZZ-function along the critical line