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Properties

Label 2-1280-20.19-c2-0-82
Degree $2$
Conductor $1280$
Sign $0.559 + 0.828i$
Analytic cond. $34.8774$
Root an. cond. $5.90571$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.13·3-s + (−4.14 + 2.79i)5-s + 6.19·7-s + 17.3·9-s − 20.0i·11-s − 15.8i·13-s + (−21.2 + 14.3i)15-s − 6.98i·17-s − 10.3i·19-s + 31.7·21-s − 22.3·23-s + (9.35 − 23.1i)25-s + 42.9·27-s + 4.20·29-s + 20.7i·31-s + ⋯
L(s)  = 1  + 1.71·3-s + (−0.828 + 0.559i)5-s + 0.884·7-s + 1.92·9-s − 1.82i·11-s − 1.22i·13-s + (−1.41 + 0.957i)15-s − 0.411i·17-s − 0.546i·19-s + 1.51·21-s − 0.973·23-s + (0.374 − 0.927i)25-s + 1.58·27-s + 0.145·29-s + 0.668i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.559 + 0.828i$
Analytic conductor: \(34.8774\)
Root analytic conductor: \(5.90571\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1),\ 0.559 + 0.828i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.339427218\)
\(L(\frac12)\) \(\approx\) \(3.339427218\)
\(L(2)\) 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 \( 1 + (4.14 - 2.79i)T \)
good3 \( 1 - 5.13T + 9T^{2} \)
7 \( 1 - 6.19T + 49T^{2} \)
11 \( 1 + 20.0iT - 121T^{2} \)
13 \( 1 + 15.8iT - 169T^{2} \)
17 \( 1 + 6.98iT - 289T^{2} \)
19 \( 1 + 10.3iT - 361T^{2} \)
23 \( 1 + 22.3T + 529T^{2} \)
29 \( 1 - 4.20T + 841T^{2} \)
31 \( 1 - 20.7iT - 961T^{2} \)
37 \( 1 + 35.4iT - 1.36e3T^{2} \)
41 \( 1 - 37.0T + 1.68e3T^{2} \)
43 \( 1 + 23.8T + 1.84e3T^{2} \)
47 \( 1 + 48.7T + 2.20e3T^{2} \)
53 \( 1 - 77.4iT - 2.80e3T^{2} \)
59 \( 1 + 0.497iT - 3.48e3T^{2} \)
61 \( 1 - 60.7T + 3.72e3T^{2} \)
67 \( 1 - 82.5T + 4.48e3T^{2} \)
71 \( 1 - 28.7iT - 5.04e3T^{2} \)
73 \( 1 + 10.1iT - 5.32e3T^{2} \)
79 \( 1 + 87.5iT - 6.24e3T^{2} \)
83 \( 1 - 103.T + 6.88e3T^{2} \)
89 \( 1 - 49.2T + 7.92e3T^{2} \)
97 \( 1 - 84.4iT - 9.40e3T^{2} \)
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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

−9.013356883478424005948563534852, −8.374826000177006493901962988988, −7.954937790901421351361692091423, −7.34870190023572470531844863613, −6.12381456727218842977784507901, −4.89174281523352057618228561454, −3.72652717950142214742652506227, −3.20206726550875008334563167908, −2.36409659470641549573859822721, −0.77679257578824010413170795607, 1.65151514975995528988271663906, 2.12590873535630487265950588451, 3.65273458343384114083893074742, 4.29844153361015710134524239646, 4.88801044470142970128769503178, 6.69391686019312208532901473519, 7.50680545540295979040833920777, 8.105057311236865927046237563921, 8.559103856992824592177558480058, 9.612984368047124787215863993686

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