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Properties

Label 2-384-8.5-c3-0-13
Degree 22
Conductor 384384
Sign 0.707+0.707i0.707 + 0.707i
Analytic cond. 22.656722.6567
Root an. cond. 4.759904.75990
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s + 8i·5-s − 12·7-s − 9·9-s − 12i·11-s + 20i·13-s + 24·15-s + 62·17-s − 108i·19-s + 36i·21-s + 72·23-s + 61·25-s + 27i·27-s − 128i·29-s + 204·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.715i·5-s − 0.647·7-s − 0.333·9-s − 0.328i·11-s + 0.426i·13-s + 0.413·15-s + 0.884·17-s − 1.30i·19-s + 0.374i·21-s + 0.652·23-s + 0.487·25-s + 0.192i·27-s − 0.819i·29-s + 1.18·31-s + ⋯

Functional equation

Λ(s)=(384s/2 ΓC(s) L(s)=((0.707+0.707i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(384s/2 ΓC(s+3/2) L(s)=((0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 384384    =    27⋅32^{7} \cdot 3
Sign: 0.707+0.707i0.707 + 0.707i
Analytic conductor: 22.656722.6567
Root analytic conductor: 4.759904.75990
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ384(193,⋅)\chi_{384} (193, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 384, ( :3/2), 0.707+0.707i)(2,\ 384,\ (\ :3/2),\ 0.707 + 0.707i)

Particular Values

L(2)L(2) ≈\approx 1.6620489241.662048924
L(12)L(\frac12) ≈\approx 1.6620489241.662048924
L(52)L(\frac{5}{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
3 1+3iT 1 + 3iT
good5 1−8iT−125T2 1 - 8iT - 125T^{2}
7 1+12T+343T2 1 + 12T + 343T^{2}
11 1+12iT−1.33e3T2 1 + 12iT - 1.33e3T^{2}
13 1−20iT−2.19e3T2 1 - 20iT - 2.19e3T^{2}
17 1−62T+4.91e3T2 1 - 62T + 4.91e3T^{2}
19 1+108iT−6.85e3T2 1 + 108iT - 6.85e3T^{2}
23 1−72T+1.21e4T2 1 - 72T + 1.21e4T^{2}
29 1+128iT−2.43e4T2 1 + 128iT - 2.43e4T^{2}
31 1−204T+2.97e4T2 1 - 204T + 2.97e4T^{2}
37 1−228iT−5.06e4T2 1 - 228iT - 5.06e4T^{2}
41 1+22T+6.89e4T2 1 + 22T + 6.89e4T^{2}
43 1+204iT−7.95e4T2 1 + 204iT - 7.95e4T^{2}
47 1−600T+1.03e5T2 1 - 600T + 1.03e5T^{2}
53 1+256iT−1.48e5T2 1 + 256iT - 1.48e5T^{2}
59 1+828iT−2.05e5T2 1 + 828iT - 2.05e5T^{2}
61 1+84iT−2.26e5T2 1 + 84iT - 2.26e5T^{2}
67 1+348iT−3.00e5T2 1 + 348iT - 3.00e5T^{2}
71 1+456T+3.57e5T2 1 + 456T + 3.57e5T^{2}
73 1−822T+3.89e5T2 1 - 822T + 3.89e5T^{2}
79 1−1.35e3T+4.93e5T2 1 - 1.35e3T + 4.93e5T^{2}
83 1+108iT−5.71e5T2 1 + 108iT - 5.71e5T^{2}
89 1+938T+7.04e5T2 1 + 938T + 7.04e5T^{2}
97 1−1.27e3T+9.12e5T2 1 - 1.27e3T + 9.12e5T^{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

−10.88758045829862777463725896634, −9.908757253579809690057150085985, −8.970381761779671909165505777745, −7.88366026597079824288528619576, −6.84660943388273585944637826379, −6.33506803100283482008421661977, −4.99321205682029885702958347931, −3.41710824678293383153661424220, −2.50033972193491986500237740914, −0.72028353492479047861093909844, 1.04797873822628727908765330350, 2.92198359829609070233142221255, 4.04303773165259767684355044715, 5.16738416771866755775375752886, 6.03029862734700909799207496381, 7.36200899198744230967914767662, 8.405451713842020603646094967365, 9.278962047227124056166872836397, 10.07581722560803102455752514939, 10.80794673550282631160108035630

Graph of the ZZ-function along the critical line