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Properties

Label 2-80-16.5-c1-0-0
Degree 22
Conductor 8080
Sign −0.943−0.332i-0.943 - 0.332i
Analytic cond. 0.6388030.638803
Root an. cond. 0.7992510.799251
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  + (−0.257 + 1.39i)2-s + (−1.66 + 1.66i)3-s + (−1.86 − 0.715i)4-s + (−0.707 − 0.707i)5-s + (−1.88 − 2.74i)6-s + 2.89i·7-s + (1.47 − 2.41i)8-s − 2.53i·9-s + (1.16 − 0.801i)10-s + (1.84 + 1.84i)11-s + (4.29 − 1.91i)12-s + (−3.08 + 3.08i)13-s + (−4.02 − 0.744i)14-s + 2.35·15-s + (2.97 + 2.67i)16-s + 7.29·17-s + ⋯
L(s)  = 1  + (−0.181 + 0.983i)2-s + (−0.960 + 0.960i)3-s + (−0.933 − 0.357i)4-s + (−0.316 − 0.316i)5-s + (−0.769 − 1.11i)6-s + 1.09i·7-s + (0.521 − 0.853i)8-s − 0.845i·9-s + (0.368 − 0.253i)10-s + (0.556 + 0.556i)11-s + (1.24 − 0.553i)12-s + (−0.854 + 0.854i)13-s + (−1.07 − 0.198i)14-s + 0.607·15-s + (0.744 + 0.667i)16-s + 1.77·17-s + ⋯

Functional equation

Λ(s)=(80s/2 ΓC(s) L(s)=((−0.943−0.332i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(80s/2 ΓC(s+1/2) L(s)=((−0.943−0.332i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8080    =    24⋅52^{4} \cdot 5
Sign: −0.943−0.332i-0.943 - 0.332i
Analytic conductor: 0.6388030.638803
Root analytic conductor: 0.7992510.799251
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ80(21,⋅)\chi_{80} (21, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 80, ( :1/2), −0.943−0.332i)(2,\ 80,\ (\ :1/2),\ -0.943 - 0.332i)

Particular Values

L(1)L(1) ≈\approx 0.0928056+0.542761i0.0928056 + 0.542761i
L(12)L(\frac12) ≈\approx 0.0928056+0.542761i0.0928056 + 0.542761i
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+(0.257−1.39i)T 1 + (0.257 - 1.39i)T
5 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
good3 1+(1.66−1.66i)T−3iT2 1 + (1.66 - 1.66i)T - 3iT^{2}
7 1−2.89iT−7T2 1 - 2.89iT - 7T^{2}
11 1+(−1.84−1.84i)T+11iT2 1 + (-1.84 - 1.84i)T + 11iT^{2}
13 1+(3.08−3.08i)T−13iT2 1 + (3.08 - 3.08i)T - 13iT^{2}
17 1−7.29T+17T2 1 - 7.29T + 17T^{2}
19 1+(1.23−1.23i)T−19iT2 1 + (1.23 - 1.23i)T - 19iT^{2}
23 1+4.60iT−23T2 1 + 4.60iT - 23T^{2}
29 1+(−4.24+4.24i)T−29iT2 1 + (-4.24 + 4.24i)T - 29iT^{2}
31 1−2.06T+31T2 1 - 2.06T + 31T^{2}
37 1+(1.17+1.17i)T+37iT2 1 + (1.17 + 1.17i)T + 37iT^{2}
41 1−4.61iT−41T2 1 - 4.61iT - 41T^{2}
43 1+(−3.03−3.03i)T+43iT2 1 + (-3.03 - 3.03i)T + 43iT^{2}
47 1+11.7T+47T2 1 + 11.7T + 47T^{2}
53 1+(−2.73−2.73i)T+53iT2 1 + (-2.73 - 2.73i)T + 53iT^{2}
59 1+(−3.11−3.11i)T+59iT2 1 + (-3.11 - 3.11i)T + 59iT^{2}
61 1+(−2.34+2.34i)T−61iT2 1 + (-2.34 + 2.34i)T - 61iT^{2}
67 1+(−8.24+8.24i)T−67iT2 1 + (-8.24 + 8.24i)T - 67iT^{2}
71 1+3.25iT−71T2 1 + 3.25iT - 71T^{2}
73 1−12.6iT−73T2 1 - 12.6iT - 73T^{2}
79 1+0.113T+79T2 1 + 0.113T + 79T^{2}
83 1+(−9.76+9.76i)T−83iT2 1 + (-9.76 + 9.76i)T - 83iT^{2}
89 1+3.74iT−89T2 1 + 3.74iT - 89T^{2}
97 1+13.9T+97T2 1 + 13.9T + 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

−15.04916599349838158583265460202, −14.37562126627049743189675909840, −12.45321322321811966929187599860, −11.77723480016359803312400669645, −10.10380359667736535198896798707, −9.403617648452073117310403902371, −8.069692462943350046187945433367, −6.43794282742513278656978756327, −5.29175108229792716368871042827, −4.35940089717309867763327357861, 0.931318203892303528770484283224, 3.47612250454271593007340283161, 5.34867073685228489067700649992, 7.04414505515548922092027906268, 8.009191967631877203615173692207, 9.902504592126109091657337519928, 10.82141117770155749271078176380, 11.81293069923267595563058990473, 12.50592131344513304143180107662, 13.57834604514510988001453272074

Graph of the ZZ-function along the critical line