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Properties

Label 2-320-20.3-c1-0-0
Degree 22
Conductor 320320
Sign 0.0898−0.995i0.0898 - 0.995i
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 + 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)=(320s/2 ΓC(s) L(s)=((0.0898−0.995i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0898 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(320s/2 ΓC(s+1/2) L(s)=((0.0898−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0898 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 320320    =    26⋅52^{6} \cdot 5
Sign: 0.0898−0.995i0.0898 - 0.995i
Analytic conductor: 2.555212.55521
Root analytic conductor: 1.598501.59850
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ320(63,⋅)\chi_{320} (63, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 320, ( :1/2), 0.0898−0.995i)(2,\ 320,\ (\ :1/2),\ 0.0898 - 0.995i)

Particular Values

L(1)L(1) ≈\approx 0.748467+0.684014i0.748467 + 0.684014i
L(12)L(\frac12) ≈\approx 0.748467+0.684014i0.748467 + 0.684014i
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

−11.45823637494045206387330414327, −11.17739061108679690381095781286, −10.21521912329344702226891179652, −8.753983348100711241549479155291, −8.203325745555624988059499181647, −7.00036321406455903231163002005, −6.17080292838263670808391925213, −4.55279357045582279894781279974, −3.73628229388653838275156087841, −2.02547983030528401194618204137, 0.76085770125354409810161120625, 3.12558842572288379734781975579, 4.12396539936214544141183389183, 5.40503843578542021326014796610, 6.59336922864393581770584593663, 7.65282297564612043595752202219, 8.674534544629272867030782860446, 9.290739722645913941443564781581, 10.75088767626022843866068533848, 11.34802083647064143476429057640

Graph of the ZZ-function along the critical line