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Properties

Label 2-80-20.3-c1-0-1
Degree 22
Conductor 8080
Sign 0.995+0.0898i0.995 + 0.0898i
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  + (2 − i)5-s + 3i·9-s + (−5 − 5i)13-s + (−5 + 5i)17-s + (3 − 4i)25-s + 4i·29-s + (5 − 5i)37-s + 8·41-s + (3 + 6i)45-s − 7i·49-s + (5 + 5i)53-s − 12·61-s + (−15 − 5i)65-s + (5 + 5i)73-s − 9·81-s + ⋯
L(s)  = 1  + (0.894 − 0.447i)5-s + i·9-s + (−1.38 − 1.38i)13-s + (−1.21 + 1.21i)17-s + (0.600 − 0.800i)25-s + 0.742i·29-s + (0.821 − 0.821i)37-s + 1.24·41-s + (0.447 + 0.894i)45-s − i·49-s + (0.686 + 0.686i)53-s − 1.53·61-s + (−1.86 − 0.620i)65-s + (0.585 + 0.585i)73-s − 81-s + ⋯

Functional equation

Λ(s)=(80s/2 ΓC(s) L(s)=((0.995+0.0898i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0898i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(80s/2 ΓC(s+1/2) L(s)=((0.995+0.0898i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0898i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8080    =    24⋅52^{4} \cdot 5
Sign: 0.995+0.0898i0.995 + 0.0898i
Analytic conductor: 0.6388030.638803
Root analytic conductor: 0.7992510.799251
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ80(63,⋅)\chi_{80} (63, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 80, ( :1/2), 0.995+0.0898i)(2,\ 80,\ (\ :1/2),\ 0.995 + 0.0898i)

Particular Values

L(1)L(1) ≈\approx 1.01291−0.0455749i1.01291 - 0.0455749i
L(12)L(\frac12) ≈\approx 1.01291−0.0455749i1.01291 - 0.0455749i
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
5 1+(−2+i)T 1 + (-2 + i)T
good3 1−3iT2 1 - 3iT^{2}
7 1+7iT2 1 + 7iT^{2}
11 1−11T2 1 - 11T^{2}
13 1+(5+5i)T+13iT2 1 + (5 + 5i)T + 13iT^{2}
17 1+(5−5i)T−17iT2 1 + (5 - 5i)T - 17iT^{2}
19 1+19T2 1 + 19T^{2}
23 1−23iT2 1 - 23iT^{2}
29 1−4iT−29T2 1 - 4iT - 29T^{2}
31 1−31T2 1 - 31T^{2}
37 1+(−5+5i)T−37iT2 1 + (-5 + 5i)T - 37iT^{2}
41 1−8T+41T2 1 - 8T + 41T^{2}
43 1−43iT2 1 - 43iT^{2}
47 1+47iT2 1 + 47iT^{2}
53 1+(−5−5i)T+53iT2 1 + (-5 - 5i)T + 53iT^{2}
59 1+59T2 1 + 59T^{2}
61 1+12T+61T2 1 + 12T + 61T^{2}
67 1+67iT2 1 + 67iT^{2}
71 1−71T2 1 - 71T^{2}
73 1+(−5−5i)T+73iT2 1 + (-5 - 5i)T + 73iT^{2}
79 1+79T2 1 + 79T^{2}
83 1−83iT2 1 - 83iT^{2}
89 1+16iT−89T2 1 + 16iT - 89T^{2}
97 1+(−5+5i)T−97iT2 1 + (-5 + 5i)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

−14.32719116835131344129120718687, −13.13304650984422156221169451965, −12.60039612390521967293181515483, −10.85059261549145160635221462498, −10.06377174762825158964122584652, −8.755222109170443434349305797563, −7.53894588705916063426304420155, −5.87142516828485734671431238577, −4.75363631493516026702961808663, −2.33847888997287159193861359754, 2.49781743409028979957278186652, 4.59227006210364649411590411646, 6.29307059904154622423476335696, 7.18411973635644872290120913485, 9.215903905955930506680129574560, 9.674845215232649046593868163176, 11.22182095132037467626831895479, 12.18903620283370482013499177600, 13.52087354184667345390636775401, 14.36196059830176043760838360071

Graph of the ZZ-function along the critical line