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Properties

Label 2-1920-12.11-c1-0-6
Degree 22
Conductor 19201920
Sign −0.934+0.356i-0.934 + 0.356i
Analytic cond. 15.331215.3312
Root an. cond. 3.915513.91551
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  + (0.618 + 1.61i)3-s + i·5-s + 2i·7-s + (−2.23 + 2.00i)9-s − 2.47·11-s − 1.23·13-s + (−1.61 + 0.618i)15-s + 0.763i·17-s + 5.23i·19-s + (−3.23 + 1.23i)21-s + 0.472·23-s − 25-s + (−4.61 − 2.38i)27-s − 8.47i·29-s − 4.76i·31-s + ⋯
L(s)  = 1  + (0.356 + 0.934i)3-s + 0.447i·5-s + 0.755i·7-s + (−0.745 + 0.666i)9-s − 0.745·11-s − 0.342·13-s + (−0.417 + 0.159i)15-s + 0.185i·17-s + 1.20i·19-s + (−0.706 + 0.269i)21-s + 0.0984·23-s − 0.200·25-s + (−0.888 − 0.458i)27-s − 1.57i·29-s − 0.855i·31-s + ⋯

Functional equation

Λ(s)=(1920s/2 ΓC(s) L(s)=((−0.934+0.356i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1920s/2 ΓC(s+1/2) L(s)=((−0.934+0.356i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19201920    =    27⋅3⋅52^{7} \cdot 3 \cdot 5
Sign: −0.934+0.356i-0.934 + 0.356i
Analytic conductor: 15.331215.3312
Root analytic conductor: 3.915513.91551
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1920(1151,⋅)\chi_{1920} (1151, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1920, ( :1/2), −0.934+0.356i)(2,\ 1920,\ (\ :1/2),\ -0.934 + 0.356i)

Particular Values

L(1)L(1) ≈\approx 0.94229077360.9422907736
L(12)L(\frac12) ≈\approx 0.94229077360.9422907736
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 1
3 1+(−0.618−1.61i)T 1 + (-0.618 - 1.61i)T
5 1−iT 1 - iT
good7 1−2iT−7T2 1 - 2iT - 7T^{2}
11 1+2.47T+11T2 1 + 2.47T + 11T^{2}
13 1+1.23T+13T2 1 + 1.23T + 13T^{2}
17 1−0.763iT−17T2 1 - 0.763iT - 17T^{2}
19 1−5.23iT−19T2 1 - 5.23iT - 19T^{2}
23 1−0.472T+23T2 1 - 0.472T + 23T^{2}
29 1+8.47iT−29T2 1 + 8.47iT - 29T^{2}
31 1+4.76iT−31T2 1 + 4.76iT - 31T^{2}
37 1+7.70T+37T2 1 + 7.70T + 37T^{2}
41 1−1.52iT−41T2 1 - 1.52iT - 41T^{2}
43 1−9.70iT−43T2 1 - 9.70iT - 43T^{2}
47 1−4.47T+47T2 1 - 4.47T + 47T^{2}
53 1+4.47iT−53T2 1 + 4.47iT - 53T^{2}
59 1+6.47T+59T2 1 + 6.47T + 59T^{2}
61 1+12.4T+61T2 1 + 12.4T + 61T^{2}
67 1−11.2iT−67T2 1 - 11.2iT - 67T^{2}
71 1−4T+71T2 1 - 4T + 71T^{2}
73 1+0.472T+73T2 1 + 0.472T + 73T^{2}
79 1+8.18iT−79T2 1 + 8.18iT - 79T^{2}
83 1−11.7T+83T2 1 - 11.7T + 83T^{2}
89 1+1.52iT−89T2 1 + 1.52iT - 89T^{2}
97 1+12.4T+97T2 1 + 12.4T + 97T^{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

−9.731013252674235107224384292080, −8.936021981826733656648280736208, −8.069794335266073431637470839999, −7.60877675813312021160365898275, −6.20289007885092285986903233434, −5.64609920965876104981226597546, −4.73109099786135824089141930353, −3.81462981556797426286736285475, −2.86556133120541789345981876709, −2.10627793849199155826054872299, 0.30949667441041139373548064271, 1.50540833196602630401450582519, 2.66240892412814673663876096987, 3.56218470095438494251460000964, 4.81451981152769045946451468919, 5.47071565102309374667598200922, 6.71450112736747976007333838852, 7.19919967409669767220143278140, 7.85505679417149756991644231506, 8.833784407893177634438954918430

Graph of the ZZ-function along the critical line