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Properties

Label 2-1280-16.13-c1-0-11
Degree $2$
Conductor $1280$
Sign $0.382 + 0.923i$
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  + (−2.21 − 2.21i)3-s + (0.707 − 0.707i)5-s − 0.275i·7-s + 6.85i·9-s + (4.13 − 4.13i)11-s + (2.71 + 2.71i)13-s − 3.13·15-s + 1.61·17-s + (1.71 + 1.71i)19-s + (−0.610 + 0.610i)21-s + 6.99i·23-s − 1.00i·25-s + (8.55 − 8.55i)27-s + (5.27 + 5.27i)29-s + 8.82·31-s + ⋯
L(s)  = 1  + (−1.28 − 1.28i)3-s + (0.316 − 0.316i)5-s − 0.104i·7-s + 2.28i·9-s + (1.24 − 1.24i)11-s + (0.752 + 0.752i)13-s − 0.810·15-s + 0.390·17-s + (0.393 + 0.393i)19-s + (−0.133 + 0.133i)21-s + 1.45i·23-s − 0.200i·25-s + (1.64 − 1.64i)27-s + (0.980 + 0.980i)29-s + 1.58·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.382 + 0.923i$
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: \((2,\ 1280,\ (\ :1/2),\ 0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.303996063\)
\(L(\frac12)\) \(\approx\) \(1.303996063\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 + 2.21i)T + 3iT^{2} \)
7 \( 1 + 0.275iT - 7T^{2} \)
11 \( 1 + (-4.13 + 4.13i)T - 11iT^{2} \)
13 \( 1 + (-2.71 - 2.71i)T + 13iT^{2} \)
17 \( 1 - 1.61T + 17T^{2} \)
19 \( 1 + (-1.71 - 1.71i)T + 19iT^{2} \)
23 \( 1 - 6.99iT - 23T^{2} \)
29 \( 1 + (-5.27 - 5.27i)T + 29iT^{2} \)
31 \( 1 - 8.82T + 31T^{2} \)
37 \( 1 + (-1.85 + 1.85i)T - 37iT^{2} \)
41 \( 1 - 5.85iT - 41T^{2} \)
43 \( 1 + (0.333 - 0.333i)T - 43iT^{2} \)
47 \( 1 + 3.10T + 47T^{2} \)
53 \( 1 + (-3.10 + 3.10i)T - 53iT^{2} \)
59 \( 1 + (3.99 - 3.99i)T - 59iT^{2} \)
61 \( 1 + (7.88 + 7.88i)T + 61iT^{2} \)
67 \( 1 + (7.60 + 7.60i)T + 67iT^{2} \)
71 \( 1 + 2.57iT - 71T^{2} \)
73 \( 1 + 3.88iT - 73T^{2} \)
79 \( 1 - 2.22T + 79T^{2} \)
83 \( 1 + (9.55 + 9.55i)T + 83iT^{2} \)
89 \( 1 - 7.62iT - 89T^{2} \)
97 \( 1 + 12.3T + 97T^{2} \)
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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

−9.475392197044213736857376665111, −8.579951848674893227836350371729, −7.77158785333863600434797018107, −6.77101828738898954896377378855, −6.20274761088172799233872254177, −5.68344340513070767915984947825, −4.60796486008307613769061257924, −3.29378198766630426531027071542, −1.52906718013386617819333026101, −1.02134191042172874961762549203, 0.964216850251806192688467743250, 2.85462496016808406253768480320, 4.15435453712189052849385099207, 4.57422485106132135242389473459, 5.67425536915173140340069049019, 6.31407705242049333756995147272, 6.99904246643489887132957924103, 8.437133394195580911960228977783, 9.325203459967043065442890136800, 10.10136170493433293731306031292

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