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Properties

Label 2-1280-8.3-c2-0-45
Degree $2$
Conductor $1280$
Sign $0.707 + 0.707i$
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  + 2.35·3-s − 2.23i·5-s − 5.25i·7-s − 3.47·9-s + 19.9·11-s + 8.47i·13-s − 5.25i·15-s + 11.8·17-s + 15.2·19-s − 12.3i·21-s − 0.555i·23-s − 5.00·25-s − 29.3·27-s + 10.9i·29-s + 8.29i·31-s + ⋯
L(s)  = 1  + 0.783·3-s − 0.447i·5-s − 0.751i·7-s − 0.385·9-s + 1.81·11-s + 0.651i·13-s − 0.350i·15-s + 0.699·17-s + 0.800·19-s − 0.588i·21-s − 0.0241i·23-s − 0.200·25-s − 1.08·27-s + 0.377i·29-s + 0.267i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(34.8774\)
Root analytic conductor: \(5.90571\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.850122505\)
\(L(\frac12)\) \(\approx\) \(2.850122505\)
\(L(2)\) 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 \( 1 + 2.23iT \)
good3 \( 1 - 2.35T + 9T^{2} \)
7 \( 1 + 5.25iT - 49T^{2} \)
11 \( 1 - 19.9T + 121T^{2} \)
13 \( 1 - 8.47iT - 169T^{2} \)
17 \( 1 - 11.8T + 289T^{2} \)
19 \( 1 - 15.2T + 361T^{2} \)
23 \( 1 + 0.555iT - 529T^{2} \)
29 \( 1 - 10.9iT - 841T^{2} \)
31 \( 1 - 8.29iT - 961T^{2} \)
37 \( 1 + 18.3iT - 1.36e3T^{2} \)
41 \( 1 - 14.5T + 1.68e3T^{2} \)
43 \( 1 - 22.2T + 1.84e3T^{2} \)
47 \( 1 + 53.3iT - 2.20e3T^{2} \)
53 \( 1 + 66.3iT - 2.80e3T^{2} \)
59 \( 1 - 17.4T + 3.48e3T^{2} \)
61 \( 1 + 90.1iT - 3.72e3T^{2} \)
67 \( 1 + 50.2T + 4.48e3T^{2} \)
71 \( 1 + 80.7iT - 5.04e3T^{2} \)
73 \( 1 - 5.55T + 5.32e3T^{2} \)
79 \( 1 - 13.8iT - 6.24e3T^{2} \)
83 \( 1 - 76.2T + 6.88e3T^{2} \)
89 \( 1 - 111.T + 7.92e3T^{2} \)
97 \( 1 + 92.8T + 9.40e3T^{2} \)
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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

−9.242990916577047490625418417732, −8.745825064410035070914046083100, −7.79620511141348608510179532660, −7.02991846889600783868580076035, −6.15336690569537299978885392767, −5.03531915710642399900059495847, −3.90397800395657631795747517653, −3.44279817825870775667512526616, −1.93962637407056010237717449222, −0.880375176487838053912375316388, 1.20595767169820111504450443387, 2.55479733187826001259582787702, 3.27535412236647002132441948565, 4.18480948198254699152892282874, 5.61307705225887120657218951357, 6.15212991469106201607802342197, 7.27915184992154283320760881358, 8.003579968317992724723110746745, 8.936434472541810629100608593569, 9.329992427519116732100412970726

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