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Properties

Label 2-1280-80.69-c1-0-10
Degree $2$
Conductor $1280$
Sign $-0.586 - 0.810i$
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  + (0.796 + 0.796i)3-s + (−1.17 + 1.90i)5-s − 0.582·7-s − 1.73i·9-s + (2.96 + 2.96i)11-s + (1.12 + 1.12i)13-s + (−2.44 + 0.582i)15-s + 5.32i·17-s + (−0.517 + 0.517i)19-s + (−0.464 − 0.464i)21-s − 6.52·23-s + (−2.25 − 4.46i)25-s + (3.76 − 3.76i)27-s + (−2.46 + 2.46i)29-s + 3.86·31-s + ⋯
L(s)  = 1  + (0.459 + 0.459i)3-s + (−0.524 + 0.851i)5-s − 0.220·7-s − 0.577i·9-s + (0.894 + 0.894i)11-s + (0.312 + 0.312i)13-s + (−0.632 + 0.150i)15-s + 1.29i·17-s + (−0.118 + 0.118i)19-s + (−0.101 − 0.101i)21-s − 1.36·23-s + (−0.450 − 0.892i)25-s + (0.725 − 0.725i)27-s + (−0.457 + 0.457i)29-s + 0.693·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.586 - 0.810i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ -0.586 - 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.427833138\)
\(L(\frac12)\) \(\approx\) \(1.427833138\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.17 - 1.90i)T \)
good3 \( 1 + (-0.796 - 0.796i)T + 3iT^{2} \)
7 \( 1 + 0.582T + 7T^{2} \)
11 \( 1 + (-2.96 - 2.96i)T + 11iT^{2} \)
13 \( 1 + (-1.12 - 1.12i)T + 13iT^{2} \)
17 \( 1 - 5.32iT - 17T^{2} \)
19 \( 1 + (0.517 - 0.517i)T - 19iT^{2} \)
23 \( 1 + 6.52T + 23T^{2} \)
29 \( 1 + (2.46 - 2.46i)T - 29iT^{2} \)
31 \( 1 - 3.86T + 31T^{2} \)
37 \( 1 + (3.90 - 3.90i)T - 37iT^{2} \)
41 \( 1 - 6.73iT - 41T^{2} \)
43 \( 1 + (-5.72 + 5.72i)T - 43iT^{2} \)
47 \( 1 - 8.11iT - 47T^{2} \)
53 \( 1 + (6.45 - 6.45i)T - 53iT^{2} \)
59 \( 1 + (2.31 + 2.31i)T + 59iT^{2} \)
61 \( 1 + (2 - 2i)T - 61iT^{2} \)
67 \( 1 + (-10.5 - 10.5i)T + 67iT^{2} \)
71 \( 1 + 13.3iT - 71T^{2} \)
73 \( 1 + 9.22T + 73T^{2} \)
79 \( 1 + 1.03T + 79T^{2} \)
83 \( 1 + (4.56 + 4.56i)T + 83iT^{2} \)
89 \( 1 + 0.535iT - 89T^{2} \)
97 \( 1 - 12.0iT - 97T^{2} \)
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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

−9.953301506710046526253888684205, −9.224415769659923178803309983685, −8.367468924077902727304620505761, −7.56274506442818244373683175159, −6.49061402911703822959790531466, −6.18333169859282351228581193032, −4.40430360807502762255123037449, −3.90855362912486009721781817601, −3.07203624402547627650934873388, −1.71237504419210754401055521514, 0.56722738055363655867622898937, 1.90226432463866484184710890899, 3.19633581047915547890414778473, 4.11664369892328014087180287742, 5.13892781741813552338080981467, 6.04395813551789370520325429044, 7.10026037054992109998109367727, 7.903353769290072001011923518367, 8.505311812911173393261132022148, 9.171764324013890235794838062190

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