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Properties

Label 2-768-8.5-c3-0-44
Degree 22
Conductor 768768
Sign −0.707−0.707i-0.707 - 0.707i
Analytic cond. 45.313445.3134
Root an. cond. 6.731526.73152
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 − 14i·5-s − 24·7-s − 9·9-s − 28i·11-s − 74i·13-s − 42·15-s + 82·17-s − 92i·19-s + 72i·21-s + 8·23-s − 71·25-s + 27i·27-s − 138i·29-s − 80·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.25i·5-s − 1.29·7-s − 0.333·9-s − 0.767i·11-s − 1.57i·13-s − 0.722·15-s + 1.16·17-s − 1.11i·19-s + 0.748i·21-s + 0.0725·23-s − 0.568·25-s + 0.192i·27-s − 0.883i·29-s − 0.463·31-s + ⋯

Functional equation

Λ(s)=(768s/2 ΓC(s) L(s)=((−0.707−0.707i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(768s/2 ΓC(s+3/2) L(s)=((−0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{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: 768768    =    28⋅32^{8} \cdot 3
Sign: −0.707−0.707i-0.707 - 0.707i
Analytic conductor: 45.313445.3134
Root analytic conductor: 6.731526.73152
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ768(385,⋅)\chi_{768} (385, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 768, ( :3/2), −0.707−0.707i)(2,\ 768,\ (\ :3/2),\ -0.707 - 0.707i)

Particular Values

L(2)L(2) ≈\approx 1.0298834471.029883447
L(12)L(\frac12) ≈\approx 1.0298834471.029883447
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+14iT−125T2 1 + 14iT - 125T^{2}
7 1+24T+343T2 1 + 24T + 343T^{2}
11 1+28iT−1.33e3T2 1 + 28iT - 1.33e3T^{2}
13 1+74iT−2.19e3T2 1 + 74iT - 2.19e3T^{2}
17 1−82T+4.91e3T2 1 - 82T + 4.91e3T^{2}
19 1+92iT−6.85e3T2 1 + 92iT - 6.85e3T^{2}
23 1−8T+1.21e4T2 1 - 8T + 1.21e4T^{2}
29 1+138iT−2.43e4T2 1 + 138iT - 2.43e4T^{2}
31 1+80T+2.97e4T2 1 + 80T + 2.97e4T^{2}
37 1+30iT−5.06e4T2 1 + 30iT - 5.06e4T^{2}
41 1+282T+6.89e4T2 1 + 282T + 6.89e4T^{2}
43 1−4iT−7.95e4T2 1 - 4iT - 7.95e4T^{2}
47 1+240T+1.03e5T2 1 + 240T + 1.03e5T^{2}
53 1−130iT−1.48e5T2 1 - 130iT - 1.48e5T^{2}
59 1−596iT−2.05e5T2 1 - 596iT - 2.05e5T^{2}
61 1+218iT−2.26e5T2 1 + 218iT - 2.26e5T^{2}
67 1−436iT−3.00e5T2 1 - 436iT - 3.00e5T^{2}
71 1−856T+3.57e5T2 1 - 856T + 3.57e5T^{2}
73 1−998T+3.89e5T2 1 - 998T + 3.89e5T^{2}
79 1−32T+4.93e5T2 1 - 32T + 4.93e5T^{2}
83 1−1.50e3iT−5.71e5T2 1 - 1.50e3iT - 5.71e5T^{2}
89 1−246T+7.04e5T2 1 - 246T + 7.04e5T^{2}
97 1−866T+9.12e5T2 1 - 866T + 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

−9.357980000937952687519437417531, −8.468018738034087619685244699889, −7.83562745865748635318416457442, −6.71365559187858951590043620734, −5.74551550750571376064039165237, −5.15055055744073820492781780992, −3.61171907099952536160179163117, −2.76678432207595766815678875213, −0.984356794191576457958615555067, −0.33227264278853253182183223493, 1.92615398707818483023844590106, 3.25465306421631206924755681275, 3.73665523670741735984778100129, 5.09051760507480844054882651891, 6.36488920915821683270876777282, 6.76412950236055971932347983173, 7.73507435079042067658638917298, 9.062189721558620557559479501246, 9.843096620261683063123788637639, 10.19163377475394451099004904742

Graph of the ZZ-function along the critical line