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Properties

Label 2-1280-8.5-c3-0-3
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  + 4.47i·3-s − 5i·5-s − 31.3·7-s + 6.99·9-s + 8.94i·11-s + 62i·13-s + 22.3·15-s − 46·17-s + 107. i·19-s − 140i·21-s + 192.·23-s − 25·25-s + 152. i·27-s + 90i·29-s + 152.·31-s + ⋯
L(s)  = 1  + 0.860i·3-s − 0.447i·5-s − 1.69·7-s + 0.259·9-s + 0.245i·11-s + 1.32i·13-s + 0.384·15-s − 0.656·17-s + 1.29i·19-s − 1.45i·21-s + 1.74·23-s − 0.200·25-s + 1.08i·27-s + 0.576i·29-s + 0.880·31-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

\(L(2)\) \(\approx\) \(0.3761445709\)
\(L(\frac12)\) \(\approx\) \(0.3761445709\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.47iT - 27T^{2} \)
7 \( 1 + 31.3T + 343T^{2} \)
11 \( 1 - 8.94iT - 1.33e3T^{2} \)
13 \( 1 - 62iT - 2.19e3T^{2} \)
17 \( 1 + 46T + 4.91e3T^{2} \)
19 \( 1 - 107. iT - 6.85e3T^{2} \)
23 \( 1 - 192.T + 1.21e4T^{2} \)
29 \( 1 - 90iT - 2.43e4T^{2} \)
31 \( 1 - 152.T + 2.97e4T^{2} \)
37 \( 1 + 214iT - 5.06e4T^{2} \)
41 \( 1 - 10T + 6.89e4T^{2} \)
43 \( 1 - 67.0iT - 7.95e4T^{2} \)
47 \( 1 + 398.T + 1.03e5T^{2} \)
53 \( 1 + 678iT - 1.48e5T^{2} \)
59 \( 1 - 411. iT - 2.05e5T^{2} \)
61 \( 1 + 250iT - 2.26e5T^{2} \)
67 \( 1 - 49.1iT - 3.00e5T^{2} \)
71 \( 1 + 366.T + 3.57e5T^{2} \)
73 \( 1 + 522T + 3.89e5T^{2} \)
79 \( 1 + 876.T + 4.93e5T^{2} \)
83 \( 1 - 380. iT - 5.71e5T^{2} \)
89 \( 1 + 970T + 7.04e5T^{2} \)
97 \( 1 + 934T + 9.12e5T^{2} \)
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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

−9.640527528386427250088905975732, −9.292689749121940998907952504201, −8.490304260294749661648194961014, −7.04280366621862543695144092733, −6.66002748437146402496611828825, −5.55217778415788084856059570988, −4.52706114123846418432775803782, −3.87136145907700723040194633779, −2.95669779110963077305929985737, −1.50222520312761604120506180272, 0.10031340865744941966204890476, 1.02433737059245267682862169867, 2.77270845465095908462591800105, 3.01918008628723528847185945080, 4.44457021036536576879944503336, 5.68629735062512243416965916287, 6.65356687791172360875548144603, 6.85018687114563848646621162025, 7.80487213984748130902835536669, 8.805355208717398810455407199720

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