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Properties

Label 2-378-63.4-c1-0-7
Degree 22
Conductor 378378
Sign −0.999−0.00294i-0.999 - 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 − 1.76·5-s + (−1.85 − 1.88i)7-s − 0.999·8-s + (−0.880 + 1.52i)10-s − 6.12·11-s + (−0.380 + 0.658i)13-s + (−2.56 + 0.658i)14-s + (−0.5 + 0.866i)16-s + (3.42 − 5.92i)17-s + (0.971 + 1.68i)19-s + (0.880 + 1.52i)20-s + (−3.06 + 5.30i)22-s + 0.421·23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s − 0.787·5-s + (−0.699 − 0.714i)7-s − 0.353·8-s + (−0.278 + 0.482i)10-s − 1.84·11-s + (−0.105 + 0.182i)13-s + (−0.684 + 0.176i)14-s + (−0.125 + 0.216i)16-s + (0.829 − 1.43i)17-s + (0.222 + 0.385i)19-s + (0.196 + 0.340i)20-s + (−0.652 + 1.13i)22-s + 0.0877·23-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: −0.999−0.00294i-0.999 - 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 0.000903517+0.613459i0.000903517 + 0.613459i
L(12)L(\frac12) ≈\approx 0.000903517+0.613459i0.000903517 + 0.613459i
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+(1.85+1.88i)T 1 + (1.85 + 1.88i)T
good5 1+1.76T+5T2 1 + 1.76T + 5T^{2}
11 1+6.12T+11T2 1 + 6.12T + 11T^{2}
13 1+(0.380−0.658i)T+(−6.5−11.2i)T2 1 + (0.380 - 0.658i)T + (-6.5 - 11.2i)T^{2}
17 1+(−3.42+5.92i)T+(−8.5−14.7i)T2 1 + (-3.42 + 5.92i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.971−1.68i)T+(−9.5+16.4i)T2 1 + (-0.971 - 1.68i)T + (-9.5 + 16.4i)T^{2}
23 1−0.421T+23T2 1 - 0.421T + 23T^{2}
29 1+(0.732+1.26i)T+(−14.5+25.1i)T2 1 + (0.732 + 1.26i)T + (-14.5 + 25.1i)T^{2}
31 1+(3.85+6.67i)T+(−15.5+26.8i)T2 1 + (3.85 + 6.67i)T + (-15.5 + 26.8i)T^{2}
37 1+(−1.44−2.49i)T+(−18.5+32.0i)T2 1 + (-1.44 - 2.49i)T + (-18.5 + 32.0i)T^{2}
41 1+(−3.47+6.01i)T+(−20.5−35.5i)T2 1 + (-3.47 + 6.01i)T + (-20.5 - 35.5i)T^{2}
43 1+(−4.33−7.49i)T+(−21.5+37.2i)T2 1 + (-4.33 - 7.49i)T + (-21.5 + 37.2i)T^{2}
47 1+(−0.830+1.43i)T+(−23.5−40.7i)T2 1 + (-0.830 + 1.43i)T + (-23.5 - 40.7i)T^{2}
53 1+(−0.112+0.195i)T+(−26.5−45.8i)T2 1 + (-0.112 + 0.195i)T + (-26.5 - 45.8i)T^{2}
59 1+(−0.993−1.72i)T+(−29.5+51.0i)T2 1 + (-0.993 - 1.72i)T + (-29.5 + 51.0i)T^{2}
61 1+(−5.17+8.96i)T+(−30.5−52.8i)T2 1 + (-5.17 + 8.96i)T + (-30.5 - 52.8i)T^{2}
67 1+(3.39+5.87i)T+(−33.5+58.0i)T2 1 + (3.39 + 5.87i)T + (-33.5 + 58.0i)T^{2}
71 1+10.7T+71T2 1 + 10.7T + 71T^{2}
73 1+(−0.153+0.265i)T+(−36.5−63.2i)T2 1 + (-0.153 + 0.265i)T + (-36.5 - 63.2i)T^{2}
79 1+(−6.72+11.6i)T+(−39.5−68.4i)T2 1 + (-6.72 + 11.6i)T + (-39.5 - 68.4i)T^{2}
83 1+(−1.56−2.70i)T+(−41.5+71.8i)T2 1 + (-1.56 - 2.70i)T + (-41.5 + 71.8i)T^{2}
89 1+(1.30+2.25i)T+(−44.5+77.0i)T2 1 + (1.30 + 2.25i)T + (-44.5 + 77.0i)T^{2}
97 1+(1.81+3.14i)T+(−48.5+84.0i)T2 1 + (1.81 + 3.14i)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

−10.97144798223153818195343124141, −10.11169026628998749026504886155, −9.406912235511257425534819574308, −7.82494211585960394911146107302, −7.40905527821513094616404060027, −5.84556050361431472095173972991, −4.79422875787954971046199739274, −3.64865809708631476687084111966, −2.65340581656796967192014953991, −0.34819613831173027322640924367, 2.73773347524747614503043618865, 3.80189995746138570002847682005, 5.22668999589650630682799591468, 5.89759429932511178320383580650, 7.24046152598763549378944065686, 7.975512004276375213294539089884, 8.774887797352642442662772820154, 10.03046998486854960710988080136, 10.88686479881959868189646208150, 12.13285063752530699729222655532

Graph of the ZZ-function along the critical line