Properties

Label 2-1280-16.13-c1-0-22
Degree $2$
Conductor $1280$
Sign $0.130 + 0.991i$
Analytic cond. $10.2208$
Root an. cond. $3.19700$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.658 + 0.658i)3-s + (−0.707 + 0.707i)5-s − 2.34i·7-s − 2.13i·9-s + (−1.19 + 1.19i)11-s + (−1.51 − 1.51i)13-s − 0.931·15-s + 1.11·17-s + (1.53 + 1.53i)19-s + (1.54 − 1.54i)21-s − 4.83i·23-s − 1.00i·25-s + (3.38 − 3.38i)27-s + (−5.49 − 5.49i)29-s − 6.29·31-s + ⋯
L(s)  = 1  + (0.380 + 0.380i)3-s + (−0.316 + 0.316i)5-s − 0.886i·7-s − 0.710i·9-s + (−0.361 + 0.361i)11-s + (−0.420 − 0.420i)13-s − 0.240·15-s + 0.269·17-s + (0.351 + 0.351i)19-s + (0.337 − 0.337i)21-s − 1.00i·23-s − 0.200i·25-s + (0.650 − 0.650i)27-s + (−1.02 − 1.02i)29-s − 1.13·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.130 + 0.991i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 0.130 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.277890266\)
\(L(\frac12)\) \(\approx\) \(1.277890266\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (-0.658 - 0.658i)T + 3iT^{2} \)
7 \( 1 + 2.34iT - 7T^{2} \)
11 \( 1 + (1.19 - 1.19i)T - 11iT^{2} \)
13 \( 1 + (1.51 + 1.51i)T + 13iT^{2} \)
17 \( 1 - 1.11T + 17T^{2} \)
19 \( 1 + (-1.53 - 1.53i)T + 19iT^{2} \)
23 \( 1 + 4.83iT - 23T^{2} \)
29 \( 1 + (5.49 + 5.49i)T + 29iT^{2} \)
31 \( 1 + 6.29T + 31T^{2} \)
37 \( 1 + (-4.59 + 4.59i)T - 37iT^{2} \)
41 \( 1 + 10.0iT - 41T^{2} \)
43 \( 1 + (5.14 - 5.14i)T - 43iT^{2} \)
47 \( 1 + 6.44T + 47T^{2} \)
53 \( 1 + (-5.02 + 5.02i)T - 53iT^{2} \)
59 \( 1 + (-1.46 + 1.46i)T - 59iT^{2} \)
61 \( 1 + (0.752 + 0.752i)T + 61iT^{2} \)
67 \( 1 + (-11.2 - 11.2i)T + 67iT^{2} \)
71 \( 1 - 0.399iT - 71T^{2} \)
73 \( 1 + 6.02iT - 73T^{2} \)
79 \( 1 - 15.8T + 79T^{2} \)
83 \( 1 + (0.312 + 0.312i)T + 83iT^{2} \)
89 \( 1 + 16.5iT - 89T^{2} \)
97 \( 1 - 9.14T + 97T^{2} \)
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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

−9.668524696328281693475201538844, −8.692889586900787682002533441643, −7.74050920500428196989365570816, −7.21796004673744827292118940983, −6.23764865948710342710770456059, −5.15015188690804895066976133164, −4.04382309357680726778837496853, −3.51674136656062556999430908351, −2.29997383530727751574418652995, −0.50416099496102841068743709117, 1.57278802539861002893849014356, 2.61295314838822223846381872278, 3.60518852533683823721140524862, 5.03275363409467060113195231199, 5.42438136283792955630656741694, 6.68971854778275149820125284876, 7.61752700522388931111259380885, 8.132580133635907552210321667704, 9.039580771277452365432971600176, 9.585527372452344252222743508441

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