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Properties

Label 2-320-16.13-c1-0-3
Degree 22
Conductor 320320
Sign 0.557−0.830i0.557 - 0.830i
Analytic cond. 2.555212.55521
Root an. cond. 1.598501.59850
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  + (2.32 + 2.32i)3-s + (0.707 − 0.707i)5-s − 0.982i·7-s + 7.82i·9-s + (1.62 − 1.62i)11-s + (−0.690 − 0.690i)13-s + 3.28·15-s − 2.19·17-s + (−1.92 − 1.92i)19-s + (2.28 − 2.28i)21-s + 2.01i·23-s − 1.00i·25-s + (−11.2 + 11.2i)27-s + (−5.27 − 5.27i)29-s − 0.435·31-s + ⋯
L(s)  = 1  + (1.34 + 1.34i)3-s + (0.316 − 0.316i)5-s − 0.371i·7-s + 2.60i·9-s + (0.490 − 0.490i)11-s + (−0.191 − 0.191i)13-s + 0.849·15-s − 0.532·17-s + (−0.441 − 0.441i)19-s + (0.498 − 0.498i)21-s + 0.420i·23-s − 0.200i·25-s + (−2.15 + 2.15i)27-s + (−0.978 − 0.978i)29-s − 0.0781·31-s + ⋯

Functional equation

Λ(s)=(320s/2 ΓC(s) L(s)=((0.557−0.830i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(320s/2 ΓC(s+1/2) L(s)=((0.557−0.830i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 320320    =    26⋅52^{6} \cdot 5
Sign: 0.557−0.830i0.557 - 0.830i
Analytic conductor: 2.555212.55521
Root analytic conductor: 1.598501.59850
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ320(81,⋅)\chi_{320} (81, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 320, ( :1/2), 0.557−0.830i)(2,\ 320,\ (\ :1/2),\ 0.557 - 0.830i)

Particular Values

L(1)L(1) ≈\approx 1.78166+0.950157i1.78166 + 0.950157i
L(12)L(\frac12) ≈\approx 1.78166+0.950157i1.78166 + 0.950157i
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+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
good3 1+(−2.32−2.32i)T+3iT2 1 + (-2.32 - 2.32i)T + 3iT^{2}
7 1+0.982iT−7T2 1 + 0.982iT - 7T^{2}
11 1+(−1.62+1.62i)T−11iT2 1 + (-1.62 + 1.62i)T - 11iT^{2}
13 1+(0.690+0.690i)T+13iT2 1 + (0.690 + 0.690i)T + 13iT^{2}
17 1+2.19T+17T2 1 + 2.19T + 17T^{2}
19 1+(1.92+1.92i)T+19iT2 1 + (1.92 + 1.92i)T + 19iT^{2}
23 1−2.01iT−23T2 1 - 2.01iT - 23T^{2}
29 1+(5.27+5.27i)T+29iT2 1 + (5.27 + 5.27i)T + 29iT^{2}
31 1+0.435T+31T2 1 + 0.435T + 31T^{2}
37 1+(5.79−5.79i)T−37iT2 1 + (5.79 - 5.79i)T - 37iT^{2}
41 1+3.93iT−41T2 1 + 3.93iT - 41T^{2}
43 1+(−0.507+0.507i)T−43iT2 1 + (-0.507 + 0.507i)T - 43iT^{2}
47 1−9.21T+47T2 1 - 9.21T + 47T^{2}
53 1+(−6.29+6.29i)T−53iT2 1 + (-6.29 + 6.29i)T - 53iT^{2}
59 1+(−5.67+5.67i)T−59iT2 1 + (-5.67 + 5.67i)T - 59iT^{2}
61 1+(3.60+3.60i)T+61iT2 1 + (3.60 + 3.60i)T + 61iT^{2}
67 1+(4.53+4.53i)T+67iT2 1 + (4.53 + 4.53i)T + 67iT^{2}
71 1−10.3iT−71T2 1 - 10.3iT - 71T^{2}
73 1−9.24iT−73T2 1 - 9.24iT - 73T^{2}
79 1+15.4T+79T2 1 + 15.4T + 79T^{2}
83 1+(−0.683−0.683i)T+83iT2 1 + (-0.683 - 0.683i)T + 83iT^{2}
89 1+5.44iT−89T2 1 + 5.44iT - 89T^{2}
97 1−5.54T+97T2 1 - 5.54T + 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

−11.50469487829456717190338453447, −10.55229597552234797182944133633, −9.812231921978649597133476114656, −8.974876629632275636956181151706, −8.401746163462147500016589119775, −7.21607424936166234516508225610, −5.55574143730971205646307758081, −4.40912153767375379151585791299, −3.58901982868440714644626810033, −2.26847662515630972629507063813, 1.70961435925483631530039717175, 2.64006209011260313467411738457, 3.95974038263984441374759071592, 5.91600465655867296964484728964, 6.93817077339998694128270177824, 7.52368578910402500039656226620, 8.814604844581719509380062168749, 9.127912757250589837672882681093, 10.45821353937183284835655903518, 11.88337152405043553905585572262

Graph of the ZZ-function along the critical line