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Properties

Label 2-1280-40.19-c2-0-22
Degree $2$
Conductor $1280$
Sign $-0.741 - 0.671i$
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  + 2.00i·3-s + (4.99 − 0.247i)5-s − 4.85·7-s + 4.98·9-s − 0.0522·11-s − 8.88·13-s + (0.496 + 10.0i)15-s + 3.74i·17-s − 20.3·19-s − 9.73i·21-s − 15.3·23-s + (24.8 − 2.47i)25-s + 28.0i·27-s + 38.3i·29-s + 34.5i·31-s + ⋯
L(s)  = 1  + 0.668i·3-s + (0.998 − 0.0495i)5-s − 0.693·7-s + 0.553·9-s − 0.00475·11-s − 0.683·13-s + (0.0331 + 0.667i)15-s + 0.220i·17-s − 1.07·19-s − 0.463i·21-s − 0.668·23-s + (0.995 − 0.0990i)25-s + 1.03i·27-s + 1.32i·29-s + 1.11i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.741 - 0.671i$
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: \((2,\ 1280,\ (\ :1),\ -0.741 - 0.671i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.409634203\)
\(L(\frac12)\) \(\approx\) \(1.409634203\)
\(L(2)\) 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 \( 1 + (-4.99 + 0.247i)T \)
good3 \( 1 - 2.00iT - 9T^{2} \)
7 \( 1 + 4.85T + 49T^{2} \)
11 \( 1 + 0.0522T + 121T^{2} \)
13 \( 1 + 8.88T + 169T^{2} \)
17 \( 1 - 3.74iT - 289T^{2} \)
19 \( 1 + 20.3T + 361T^{2} \)
23 \( 1 + 15.3T + 529T^{2} \)
29 \( 1 - 38.3iT - 841T^{2} \)
31 \( 1 - 34.5iT - 961T^{2} \)
37 \( 1 + 13.5T + 1.36e3T^{2} \)
41 \( 1 - 30.8T + 1.68e3T^{2} \)
43 \( 1 - 5.41iT - 1.84e3T^{2} \)
47 \( 1 + 59.9T + 2.20e3T^{2} \)
53 \( 1 - 5.15T + 2.80e3T^{2} \)
59 \( 1 - 88.9T + 3.48e3T^{2} \)
61 \( 1 - 90.8iT - 3.72e3T^{2} \)
67 \( 1 - 71.2iT - 4.48e3T^{2} \)
71 \( 1 + 125. iT - 5.04e3T^{2} \)
73 \( 1 - 113. iT - 5.32e3T^{2} \)
79 \( 1 - 1.80iT - 6.24e3T^{2} \)
83 \( 1 - 135. iT - 6.88e3T^{2} \)
89 \( 1 - 70.2T + 7.92e3T^{2} \)
97 \( 1 + 106. iT - 9.40e3T^{2} \)
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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

−9.920310139784999830586440830850, −9.166356859096642860564524632466, −8.423635537043612960218773840249, −7.08541349859433387126341372283, −6.53622992285862021137239041987, −5.51300236841996271736952198194, −4.73885709656581640784954030046, −3.75409270147754021481484423428, −2.65652377428096658270202418562, −1.49953907360576819994084500594, 0.38159960096338841005499973910, 1.87139191806478487426896535657, 2.53711092624204571493152062605, 3.94765407459336501460620898224, 4.99283407743680463530139882379, 6.19084366647745967406042274377, 6.47741894239440241860789668790, 7.46307358666467650713666814091, 8.254611969742556864639710185166, 9.453959892544459513089812924868

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