Properties

Label 2-1280-8.5-c3-0-56
Degree $2$
Conductor $1280$
Sign $0.707 + 0.707i$
Analytic cond. $75.5224$
Root an. cond. $8.69036$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6i·3-s + 5i·5-s + 34·7-s − 9·9-s − 16i·11-s + 58i·13-s + 30·15-s − 70·17-s + 4i·19-s − 204i·21-s + 134·23-s − 25·25-s − 108i·27-s − 242i·29-s + 100·31-s + ⋯
L(s)  = 1  − 1.15i·3-s + 0.447i·5-s + 1.83·7-s − 0.333·9-s − 0.438i·11-s + 1.23i·13-s + 0.516·15-s − 0.998·17-s + 0.0482i·19-s − 2.11i·21-s + 1.21·23-s − 0.200·25-s − 0.769i·27-s − 1.54i·29-s + 0.579·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(75.5224\)
Root analytic conductor: \(8.69036\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :3/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.913247195\)
\(L(\frac12)\) \(\approx\) \(2.913247195\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - 5iT \)
good3 \( 1 + 6iT - 27T^{2} \)
7 \( 1 - 34T + 343T^{2} \)
11 \( 1 + 16iT - 1.33e3T^{2} \)
13 \( 1 - 58iT - 2.19e3T^{2} \)
17 \( 1 + 70T + 4.91e3T^{2} \)
19 \( 1 - 4iT - 6.85e3T^{2} \)
23 \( 1 - 134T + 1.21e4T^{2} \)
29 \( 1 + 242iT - 2.43e4T^{2} \)
31 \( 1 - 100T + 2.97e4T^{2} \)
37 \( 1 - 438iT - 5.06e4T^{2} \)
41 \( 1 - 138T + 6.89e4T^{2} \)
43 \( 1 + 178iT - 7.95e4T^{2} \)
47 \( 1 - 22T + 1.03e5T^{2} \)
53 \( 1 + 162iT - 1.48e5T^{2} \)
59 \( 1 - 268iT - 2.05e5T^{2} \)
61 \( 1 - 250iT - 2.26e5T^{2} \)
67 \( 1 - 422iT - 3.00e5T^{2} \)
71 \( 1 - 852T + 3.57e5T^{2} \)
73 \( 1 + 306T + 3.89e5T^{2} \)
79 \( 1 + 456T + 4.93e5T^{2} \)
83 \( 1 - 434iT - 5.71e5T^{2} \)
89 \( 1 - 726T + 7.04e5T^{2} \)
97 \( 1 - 1.37e3T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.860509625885330217548384967013, −8.289037458492586022701750380463, −7.51317087682150145388882948578, −6.83969770964335512117670545130, −6.11265740625908952219023256737, −4.87016605719736532203805294925, −4.19283627668087246014027380717, −2.53649711330345309035095437828, −1.80695137662523594493475965700, −0.905588982788223636322414048754, 0.924150554144501769831617238809, 2.10082722876539966745034798149, 3.45507706425198813174685781113, 4.64415048540858359589040417274, 4.82089375557884177803052948610, 5.65119151514297760264601478650, 7.15151580727155384064562646661, 7.895395184816386569843687363197, 8.777935974706803563534767402723, 9.239100361430189529722606322297

Graph of the $Z$-function along the critical line