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Properties

Label 2-378-63.16-c1-0-1
Degree 22
Conductor 378378
Sign 0.384−0.923i0.384 - 0.923i
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 − 1.58·5-s + (−2.64 + 0.0963i)7-s + 0.999·8-s + (0.794 + 1.37i)10-s + 1.58·11-s + (2.40 + 4.16i)13-s + (1.40 + 2.24i)14-s + (−0.5 − 0.866i)16-s + (2.69 + 4.67i)17-s + (−3.54 + 6.14i)19-s + (0.794 − 1.37i)20-s + (−0.794 − 1.37i)22-s − 0.300·23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.710·5-s + (−0.999 + 0.0364i)7-s + 0.353·8-s + (0.251 + 0.434i)10-s + 0.478·11-s + (0.667 + 1.15i)13-s + (0.375 + 0.599i)14-s + (−0.125 − 0.216i)16-s + (0.654 + 1.13i)17-s + (−0.814 + 1.41i)19-s + (0.177 − 0.307i)20-s + (−0.169 − 0.293i)22-s − 0.0626·23-s + ⋯

Functional equation

Λ(s)=(378s/2 ΓC(s) L(s)=((0.384−0.923i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(378s/2 ΓC(s+1/2) L(s)=((0.384−0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 378378    =    2⋅33⋅72 \cdot 3^{3} \cdot 7
Sign: 0.384−0.923i0.384 - 0.923i
Analytic conductor: 3.018343.01834
Root analytic conductor: 1.737331.73733
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ378(289,⋅)\chi_{378} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 378, ( :1/2), 0.384−0.923i)(2,\ 378,\ (\ :1/2),\ 0.384 - 0.923i)

Particular Values

L(1)L(1) ≈\approx 0.519194+0.346064i0.519194 + 0.346064i
L(12)L(\frac12) ≈\approx 0.519194+0.346064i0.519194 + 0.346064i
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+(2.64−0.0963i)T 1 + (2.64 - 0.0963i)T
good5 1+1.58T+5T2 1 + 1.58T + 5T^{2}
11 1−1.58T+11T2 1 - 1.58T + 11T^{2}
13 1+(−2.40−4.16i)T+(−6.5+11.2i)T2 1 + (-2.40 - 4.16i)T + (-6.5 + 11.2i)T^{2}
17 1+(−2.69−4.67i)T+(−8.5+14.7i)T2 1 + (-2.69 - 4.67i)T + (-8.5 + 14.7i)T^{2}
19 1+(3.54−6.14i)T+(−9.5−16.4i)T2 1 + (3.54 - 6.14i)T + (-9.5 - 16.4i)T^{2}
23 1+0.300T+23T2 1 + 0.300T + 23T^{2}
29 1+(4.13−7.16i)T+(−14.5−25.1i)T2 1 + (4.13 - 7.16i)T + (-14.5 - 25.1i)T^{2}
31 1+(−1.35+2.34i)T+(−15.5−26.8i)T2 1 + (-1.35 + 2.34i)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+(2.93+5.08i)T+(−20.5+35.5i)T2 1 + (2.93 + 5.08i)T + (-20.5 + 35.5i)T^{2}
43 1+(0.833−1.44i)T+(−21.5−37.2i)T2 1 + (0.833 - 1.44i)T + (-21.5 - 37.2i)T^{2}
47 1+(−1.33−2.30i)T+(−23.5+40.7i)T2 1 + (-1.33 - 2.30i)T + (-23.5 + 40.7i)T^{2}
53 1+(2.44+4.23i)T+(−26.5+45.8i)T2 1 + (2.44 + 4.23i)T + (-26.5 + 45.8i)T^{2}
59 1+(−3.23+5.60i)T+(−29.5−51.0i)T2 1 + (-3.23 + 5.60i)T + (-29.5 - 51.0i)T^{2}
61 1+(−2.23−3.87i)T+(−30.5+52.8i)T2 1 + (-2.23 - 3.87i)T + (-30.5 + 52.8i)T^{2}
67 1+(−5.02+8.70i)T+(−33.5−58.0i)T2 1 + (-5.02 + 8.70i)T + (-33.5 - 58.0i)T^{2}
71 1+12.7T+71T2 1 + 12.7T + 71T^{2}
73 1+(−8.02−13.9i)T+(−36.5+63.2i)T2 1 + (-8.02 - 13.9i)T + (-36.5 + 63.2i)T^{2}
79 1+(4.19+7.26i)T+(−39.5+68.4i)T2 1 + (4.19 + 7.26i)T + (-39.5 + 68.4i)T^{2}
83 1+(1.18−2.04i)T+(−41.5−71.8i)T2 1 + (1.18 - 2.04i)T + (-41.5 - 71.8i)T^{2}
89 1+(1.60−2.78i)T+(−44.5−77.0i)T2 1 + (1.60 - 2.78i)T + (-44.5 - 77.0i)T^{2}
97 1+(−0.712+1.23i)T+(−48.5−84.0i)T2 1 + (-0.712 + 1.23i)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

−11.54317444974217256483310657108, −10.61904579019137042977631485427, −9.769661439535378098621529712381, −8.847915687339621451727062839948, −8.040453910574620024323675229422, −6.84660254686393453745340411892, −5.89381929066827786350808604127, −3.99317401508695538031094859079, −3.60050624706311344299279063644, −1.72543276160858166446234379667, 0.47917294819184509109412475082, 2.99197892063502348043752662188, 4.19174952634560981867959741388, 5.55940084735348107065927736712, 6.55167411460536584485989717777, 7.42667138031756938272507302555, 8.338357570662908724774989279574, 9.297056691059508565107549911203, 10.08938640986445483370144474859, 11.14733736667289451866259235613

Graph of the ZZ-function along the critical line