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Properties

Label 2-320-5.3-c0-0-0
Degree 22
Conductor 320320
Sign 0.850+0.525i0.850 + 0.525i
Analytic cond. 0.1597000.159700
Root an. cond. 0.3996250.399625
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·5-s − i·9-s + (1 + i)13-s + (−1 + i)17-s − 25-s + (−1 + i)37-s − 45-s + i·49-s + (−1 − i)53-s + (1 − i)65-s + (1 + i)73-s − 81-s + (1 + i)85-s + (1 − i)97-s + 2·101-s + ⋯
L(s)  = 1  − i·5-s − i·9-s + (1 + i)13-s + (−1 + i)17-s − 25-s + (−1 + i)37-s − 45-s + i·49-s + (−1 − i)53-s + (1 − i)65-s + (1 + i)73-s − 81-s + (1 + i)85-s + (1 − i)97-s + 2·101-s + ⋯

Functional equation

Λ(s)=(320s/2 ΓC(s) L(s)=((0.850+0.525i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(320s/2 ΓC(s) L(s)=((0.850+0.525i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 320320    =    26⋅52^{6} \cdot 5
Sign: 0.850+0.525i0.850 + 0.525i
Analytic conductor: 0.1597000.159700
Root analytic conductor: 0.3996250.399625
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ320(193,⋅)\chi_{320} (193, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 320, ( :0), 0.850+0.525i)(2,\ 320,\ (\ :0),\ 0.850 + 0.525i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.79098386310.7909838631
L(12)L(\frac12) ≈\approx 0.79098386310.7909838631
L(1)L(1) 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+iT 1 + iT
good3 1+iT2 1 + iT^{2}
7 1−iT2 1 - iT^{2}
11 1+T2 1 + T^{2}
13 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
17 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
19 1−T2 1 - T^{2}
23 1+iT2 1 + iT^{2}
29 1−T2 1 - T^{2}
31 1+T2 1 + T^{2}
37 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
41 1+T2 1 + T^{2}
43 1+iT2 1 + iT^{2}
47 1−iT2 1 - iT^{2}
53 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
59 1−T2 1 - T^{2}
61 1+T2 1 + T^{2}
67 1−iT2 1 - iT^{2}
71 1+T2 1 + T^{2}
73 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
79 1−T2 1 - T^{2}
83 1+iT2 1 + iT^{2}
89 1−T2 1 - T^{2}
97 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{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

−11.83342670478939600778000367090, −10.99823526752159599260567792653, −9.718829672758755078742899264270, −8.887964672811984972068635532586, −8.315241678892954349982762143012, −6.76267191062233945982580644459, −5.98658972706148741961646032438, −4.58466241709766521529310591437, −3.67342203931195572136873608633, −1.59942077213077209988873457183, 2.31263235827859086969458246725, 3.52007095924675564889294389541, 4.99971014154535995678329749748, 6.12915681482321798350687279061, 7.18830048997100790055883055519, 8.032732310269770728252854160536, 9.137370007529070708115527314309, 10.42385273045418601749797180161, 10.85084804999473395270961997714, 11.70730757930274210052070860578

Graph of the ZZ-function along the critical line