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Properties

Label 2-1280-40.19-c2-0-2
Degree $2$
Conductor $1280$
Sign $-0.968 + 0.248i$
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  + 2.75i·3-s + (2.54 − 4.30i)5-s − 3.84·7-s + 1.41·9-s − 6.19·11-s − 16.1·13-s + (11.8 + 7.01i)15-s + 5.20i·17-s + 36.2·19-s − 10.6i·21-s − 22.0·23-s + (−12.0 − 21.9i)25-s + 28.6i·27-s − 20.0i·29-s + 26.4i·31-s + ⋯
L(s)  = 1  + 0.918i·3-s + (0.509 − 0.860i)5-s − 0.549·7-s + 0.157·9-s − 0.562·11-s − 1.23·13-s + (0.789 + 0.467i)15-s + 0.306i·17-s + 1.90·19-s − 0.504i·21-s − 0.958·23-s + (−0.480 − 0.876i)25-s + 1.06i·27-s − 0.690i·29-s + 0.852i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.968 + 0.248i$
Analytic conductor: \(34.8774\)
Root analytic conductor: \(5.90571\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1),\ -0.968 + 0.248i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1518198986\)
\(L(\frac12)\) \(\approx\) \(0.1518198986\)
\(L(2)\) 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 \( 1 + (-2.54 + 4.30i)T \)
good3 \( 1 - 2.75iT - 9T^{2} \)
7 \( 1 + 3.84T + 49T^{2} \)
11 \( 1 + 6.19T + 121T^{2} \)
13 \( 1 + 16.1T + 169T^{2} \)
17 \( 1 - 5.20iT - 289T^{2} \)
19 \( 1 - 36.2T + 361T^{2} \)
23 \( 1 + 22.0T + 529T^{2} \)
29 \( 1 + 20.0iT - 841T^{2} \)
31 \( 1 - 26.4iT - 961T^{2} \)
37 \( 1 + 69.3T + 1.36e3T^{2} \)
41 \( 1 + 11.6T + 1.68e3T^{2} \)
43 \( 1 - 25.8iT - 1.84e3T^{2} \)
47 \( 1 + 66.1T + 2.20e3T^{2} \)
53 \( 1 - 39.5T + 2.80e3T^{2} \)
59 \( 1 + 27.7T + 3.48e3T^{2} \)
61 \( 1 + 54.1iT - 3.72e3T^{2} \)
67 \( 1 + 107. iT - 4.48e3T^{2} \)
71 \( 1 - 70.7iT - 5.04e3T^{2} \)
73 \( 1 - 37.4iT - 5.32e3T^{2} \)
79 \( 1 - 97.6iT - 6.24e3T^{2} \)
83 \( 1 - 126. iT - 6.88e3T^{2} \)
89 \( 1 + 133.T + 7.92e3T^{2} \)
97 \( 1 + 6.40iT - 9.40e3T^{2} \)
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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

−9.823408265875297304563527518850, −9.485594769665687857233612020525, −8.421854342453480056036969374350, −7.55886969097488604080402143549, −6.57178020514815075730638113701, −5.25165479392853325948755625773, −5.10934314656516449714911533246, −3.95239571696187477377044250560, −2.94921957299259647207245325724, −1.59095187298666488401002657101, 0.04093904014922065995778438076, 1.64772870790188416352749620370, 2.61194816498120914167932719397, 3.45797908504736640814738619136, 4.99626718053443591127854231818, 5.80836557800930757394116391351, 6.79781159908231651265527241917, 7.29370884404595845590493552039, 7.85412441531489285301328968399, 9.208071401818409780031885921074

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