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Properties

Label 2-320-80.3-c1-0-7
Degree 22
Conductor 320320
Sign 0.811+0.584i0.811 + 0.584i
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  + 2·3-s + (−2 − i)5-s + (3 − 3i)7-s + 9-s + (1 + i)11-s − 2i·13-s + (−4 − 2i)15-s + (1 − i)17-s + (3 + 3i)19-s + (6 − 6i)21-s + (1 + i)23-s + (3 + 4i)25-s − 4·27-s + (−7 + 7i)29-s − 2i·31-s + ⋯
L(s)  = 1  + 1.15·3-s + (−0.894 − 0.447i)5-s + (1.13 − 1.13i)7-s + 0.333·9-s + (0.301 + 0.301i)11-s − 0.554i·13-s + (−1.03 − 0.516i)15-s + (0.242 − 0.242i)17-s + (0.688 + 0.688i)19-s + (1.30 − 1.30i)21-s + (0.208 + 0.208i)23-s + (0.600 + 0.800i)25-s − 0.769·27-s + (−1.29 + 1.29i)29-s − 0.359i·31-s + ⋯

Functional equation

Λ(s)=(320s/2 ΓC(s) L(s)=((0.811+0.584i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(320s/2 ΓC(s+1/2) L(s)=((0.811+0.584i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 320320    =    26⋅52^{6} \cdot 5
Sign: 0.811+0.584i0.811 + 0.584i
Analytic conductor: 2.555212.55521
Root analytic conductor: 1.598501.59850
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ320(303,⋅)\chi_{320} (303, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 320, ( :1/2), 0.811+0.584i)(2,\ 320,\ (\ :1/2),\ 0.811 + 0.584i)

Particular Values

L(1)L(1) ≈\approx 1.68941−0.545382i1.68941 - 0.545382i
L(12)L(\frac12) ≈\approx 1.68941−0.545382i1.68941 - 0.545382i
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 1−2T+3T2 1 - 2T + 3T^{2}
7 1+(−3+3i)T−7iT2 1 + (-3 + 3i)T - 7iT^{2}
11 1+(−1−i)T+11iT2 1 + (-1 - i)T + 11iT^{2}
13 1+2iT−13T2 1 + 2iT - 13T^{2}
17 1+(−1+i)T−17iT2 1 + (-1 + i)T - 17iT^{2}
19 1+(−3−3i)T+19iT2 1 + (-3 - 3i)T + 19iT^{2}
23 1+(−1−i)T+23iT2 1 + (-1 - i)T + 23iT^{2}
29 1+(7−7i)T−29iT2 1 + (7 - 7i)T - 29iT^{2}
31 1+2iT−31T2 1 + 2iT - 31T^{2}
37 1−6iT−37T2 1 - 6iT - 37T^{2}
41 1+4iT−41T2 1 + 4iT - 41T^{2}
43 1+4iT−43T2 1 + 4iT - 43T^{2}
47 1+(−7−7i)T+47iT2 1 + (-7 - 7i)T + 47iT^{2}
53 1+8T+53T2 1 + 8T + 53T^{2}
59 1+(3−3i)T−59iT2 1 + (3 - 3i)T - 59iT^{2}
61 1+(1+i)T+61iT2 1 + (1 + i)T + 61iT^{2}
67 1−4iT−67T2 1 - 4iT - 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+(−3+3i)T−73iT2 1 + (-3 + 3i)T - 73iT^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1−2T+83T2 1 - 2T + 83T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1+(11−11i)T−97iT2 1 + (11 - 11i)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

−11.49078568171744158644264814125, −10.68888766525442977472543815130, −9.478182554347819723251539097819, −8.545716628297638734549363976848, −7.69993671153038351534142081142, −7.35378707532596062831670289749, −5.29173233385468064927013894560, −4.15979097804442808367545409375, −3.30341074745633033539460164973, −1.40955617353133347827143458429, 2.11640262027913904639651854572, 3.22925393380734205009349706187, 4.42050212556708163117585892987, 5.77524046318002532213843835306, 7.27628014750412082123667030193, 8.085843928995543131546044172507, 8.742467991714931292882188491978, 9.528214016195289226326838627861, 11.18527061447747119851428558245, 11.50814938353155228541648729155

Graph of the ZZ-function along the critical line