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Properties

Label 2-1920-80.27-c1-0-37
Degree 22
Conductor 19201920
Sign 0.937+0.348i0.937 + 0.348i
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  + 3-s + (2.15 + 0.609i)5-s + (−0.566 − 0.566i)7-s + 9-s + (3.64 − 3.64i)11-s − 2.74i·13-s + (2.15 + 0.609i)15-s + (2.08 + 2.08i)17-s + (−5.79 + 5.79i)19-s + (−0.566 − 0.566i)21-s + (4.28 − 4.28i)23-s + (4.25 + 2.62i)25-s + 27-s + (2.63 + 2.63i)29-s − 8.10i·31-s + ⋯
L(s)  = 1  + 0.577·3-s + (0.962 + 0.272i)5-s + (−0.214 − 0.214i)7-s + 0.333·9-s + (1.09 − 1.09i)11-s − 0.760i·13-s + (0.555 + 0.157i)15-s + (0.505 + 0.505i)17-s + (−1.32 + 1.32i)19-s + (−0.123 − 0.123i)21-s + (0.892 − 0.892i)23-s + (0.851 + 0.524i)25-s + 0.192·27-s + (0.489 + 0.489i)29-s − 1.45i·31-s + ⋯

Functional equation

Λ(s)=(1920s/2 ΓC(s) L(s)=((0.937+0.348i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1920s/2 ΓC(s+1/2) L(s)=((0.937+0.348i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19201920    =    27⋅3⋅52^{7} \cdot 3 \cdot 5
Sign: 0.937+0.348i0.937 + 0.348i
Analytic conductor: 15.331215.3312
Root analytic conductor: 3.915513.91551
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1920(1567,⋅)\chi_{1920} (1567, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1920, ( :1/2), 0.937+0.348i)(2,\ 1920,\ (\ :1/2),\ 0.937 + 0.348i)

Particular Values

L(1)L(1) ≈\approx 2.7387104712.738710471
L(12)L(\frac12) ≈\approx 2.7387104712.738710471
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−T 1 - T
5 1+(−2.15−0.609i)T 1 + (-2.15 - 0.609i)T
good7 1+(0.566+0.566i)T+7iT2 1 + (0.566 + 0.566i)T + 7iT^{2}
11 1+(−3.64+3.64i)T−11iT2 1 + (-3.64 + 3.64i)T - 11iT^{2}
13 1+2.74iT−13T2 1 + 2.74iT - 13T^{2}
17 1+(−2.08−2.08i)T+17iT2 1 + (-2.08 - 2.08i)T + 17iT^{2}
19 1+(5.79−5.79i)T−19iT2 1 + (5.79 - 5.79i)T - 19iT^{2}
23 1+(−4.28+4.28i)T−23iT2 1 + (-4.28 + 4.28i)T - 23iT^{2}
29 1+(−2.63−2.63i)T+29iT2 1 + (-2.63 - 2.63i)T + 29iT^{2}
31 1+8.10iT−31T2 1 + 8.10iT - 31T^{2}
37 1−2.28iT−37T2 1 - 2.28iT - 37T^{2}
41 1+2.27iT−41T2 1 + 2.27iT - 41T^{2}
43 1+3.06iT−43T2 1 + 3.06iT - 43T^{2}
47 1+(1.80−1.80i)T−47iT2 1 + (1.80 - 1.80i)T - 47iT^{2}
53 1+6.32T+53T2 1 + 6.32T + 53T^{2}
59 1+(5.56+5.56i)T+59iT2 1 + (5.56 + 5.56i)T + 59iT^{2}
61 1+(4.82−4.82i)T−61iT2 1 + (4.82 - 4.82i)T - 61iT^{2}
67 1−3.34iT−67T2 1 - 3.34iT - 67T^{2}
71 1−2.81T+71T2 1 - 2.81T + 71T^{2}
73 1+(−10.7−10.7i)T+73iT2 1 + (-10.7 - 10.7i)T + 73iT^{2}
79 1−12.1T+79T2 1 - 12.1T + 79T^{2}
83 1+1.97T+83T2 1 + 1.97T + 83T^{2}
89 1−10.0T+89T2 1 - 10.0T + 89T^{2}
97 1+(1.02+1.02i)T+97iT2 1 + (1.02 + 1.02i)T + 97iT^{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

−9.071671638405057761532791792606, −8.490681486967995319015040976737, −7.75212165753208838337746306935, −6.40871990996597046166700143110, −6.31719265364501913531784685201, −5.22459052811244121994563755050, −3.94304474120578797039616313244, −3.28038950991628692764715404153, −2.21408646170953087560274327144, −1.06944998434389685897470673899, 1.36521505059875924075895079591, 2.21741027652600697184903881549, 3.22288506160862634998887369400, 4.50622214861858612013395938425, 4.96237969562876782091896895462, 6.37578972967972955005246509918, 6.70879647154145302141591820547, 7.63264412738571025222826742588, 8.881663628920254533601197892258, 9.183019396545750948990456086384

Graph of the ZZ-function along the critical line