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Properties

Label 2-1280-40.19-c2-0-45
Degree $2$
Conductor $1280$
Sign $0.801 - 0.597i$
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  − 0.933i·3-s + (−0.723 + 4.94i)5-s + 6.91·7-s + 8.12·9-s + 11.8·11-s + 13.4·13-s + (4.61 + 0.675i)15-s + 28.7i·17-s + 30.1·19-s − 6.45i·21-s − 39.5·23-s + (−23.9 − 7.15i)25-s − 15.9i·27-s − 18.9i·29-s + 6.41i·31-s + ⋯
L(s)  = 1  − 0.311i·3-s + (−0.144 + 0.989i)5-s + 0.987·7-s + 0.903·9-s + 1.07·11-s + 1.03·13-s + (0.307 + 0.0450i)15-s + 1.69i·17-s + 1.58·19-s − 0.307i·21-s − 1.71·23-s + (−0.958 − 0.286i)25-s − 0.592i·27-s − 0.655i·29-s + 0.206i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.801 - 0.597i$
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: \((2,\ 1280,\ (\ :1),\ 0.801 - 0.597i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.707207806\)
\(L(\frac12)\) \(\approx\) \(2.707207806\)
\(L(2)\) 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 \( 1 + (0.723 - 4.94i)T \)
good3 \( 1 + 0.933iT - 9T^{2} \)
7 \( 1 - 6.91T + 49T^{2} \)
11 \( 1 - 11.8T + 121T^{2} \)
13 \( 1 - 13.4T + 169T^{2} \)
17 \( 1 - 28.7iT - 289T^{2} \)
19 \( 1 - 30.1T + 361T^{2} \)
23 \( 1 + 39.5T + 529T^{2} \)
29 \( 1 + 18.9iT - 841T^{2} \)
31 \( 1 - 6.41iT - 961T^{2} \)
37 \( 1 + 51.5T + 1.36e3T^{2} \)
41 \( 1 - 17.0T + 1.68e3T^{2} \)
43 \( 1 + 58.0iT - 1.84e3T^{2} \)
47 \( 1 - 48.1T + 2.20e3T^{2} \)
53 \( 1 - 4.02T + 2.80e3T^{2} \)
59 \( 1 - 45.6T + 3.48e3T^{2} \)
61 \( 1 + 37.8iT - 3.72e3T^{2} \)
67 \( 1 + 3.82iT - 4.48e3T^{2} \)
71 \( 1 - 119. iT - 5.04e3T^{2} \)
73 \( 1 + 82.9iT - 5.32e3T^{2} \)
79 \( 1 - 83.2iT - 6.24e3T^{2} \)
83 \( 1 - 115. iT - 6.88e3T^{2} \)
89 \( 1 + 54.7T + 7.92e3T^{2} \)
97 \( 1 - 51.8iT - 9.40e3T^{2} \)
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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

−9.722846250458222301330422365351, −8.530473917330049152666725827849, −7.915524167878859753180135709130, −7.10321556694136501633700706803, −6.34544386826106597778226603900, −5.56015983502478623391276152020, −3.98189253997882464395038760275, −3.77706453343417972753144417539, −2.02634101098199390291110038848, −1.28629954089590659107016804503, 0.966821965115130371514070696895, 1.68403466601171252605833687094, 3.48601384039022665212092047214, 4.33367194324751458896063765140, 4.99895370861848026400701101644, 5.87491878481400973359892044627, 7.12736193275956587686717351948, 7.78414010542050517094463367395, 8.714566477639266684164152885181, 9.354141301676563125466835962115

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