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Properties

Label 2-1280-20.3-c1-0-35
Degree $2$
Conductor $1280$
Sign $-0.340 + 0.940i$
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  + (2.18 − 2.18i)3-s + (−2.18 + 0.456i)5-s + (1.79 + 1.79i)7-s − 6.58i·9-s − 0.913i·11-s + (−1.73 − 1.73i)13-s + (−3.79 + 5.79i)15-s + (3 − 3i)17-s − 3.46·19-s + 7.84·21-s + (3.79 − 3.79i)23-s + (4.58 − 1.99i)25-s + (−7.84 − 7.84i)27-s − 5.29i·29-s − 7.58i·31-s + ⋯
L(s)  = 1  + (1.26 − 1.26i)3-s + (−0.978 + 0.204i)5-s + (0.677 + 0.677i)7-s − 2.19i·9-s − 0.275i·11-s + (−0.480 − 0.480i)13-s + (−0.978 + 1.49i)15-s + (0.727 − 0.727i)17-s − 0.794·19-s + 1.71·21-s + (0.790 − 0.790i)23-s + (0.916 − 0.399i)25-s + (−1.50 − 1.50i)27-s − 0.982i·29-s − 1.36i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.340 + 0.940i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (1023, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ -0.340 + 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.106732765\)
\(L(\frac12)\) \(\approx\) \(2.106732765\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.18 - 0.456i)T \)
good3 \( 1 + (-2.18 + 2.18i)T - 3iT^{2} \)
7 \( 1 + (-1.79 - 1.79i)T + 7iT^{2} \)
11 \( 1 + 0.913iT - 11T^{2} \)
13 \( 1 + (1.73 + 1.73i)T + 13iT^{2} \)
17 \( 1 + (-3 + 3i)T - 17iT^{2} \)
19 \( 1 + 3.46T + 19T^{2} \)
23 \( 1 + (-3.79 + 3.79i)T - 23iT^{2} \)
29 \( 1 + 5.29iT - 29T^{2} \)
31 \( 1 + 7.58iT - 31T^{2} \)
37 \( 1 + (5.19 - 5.19i)T - 37iT^{2} \)
41 \( 1 + 1.58T + 41T^{2} \)
43 \( 1 + (0.361 - 0.361i)T - 43iT^{2} \)
47 \( 1 + (-3.79 - 3.79i)T + 47iT^{2} \)
53 \( 1 + (2.64 + 2.64i)T + 53iT^{2} \)
59 \( 1 - 5.29T + 59T^{2} \)
61 \( 1 - 6.20T + 61T^{2} \)
67 \( 1 + (-0.361 - 0.361i)T + 67iT^{2} \)
71 \( 1 - 4.41iT - 71T^{2} \)
73 \( 1 + (-8.58 - 8.58i)T + 73iT^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + (-3.10 + 3.10i)T - 83iT^{2} \)
89 \( 1 - 15.1iT - 89T^{2} \)
97 \( 1 + (8.58 - 8.58i)T - 97iT^{2} \)
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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

−9.042810440542378011357861053336, −8.307210221496525243143983670314, −7.976122988161769952035980937145, −7.20779144128685822563524358987, −6.43848757059042342371469713838, −5.18396030178573080996403696294, −3.96169076421326649443595374970, −2.89419499199611230187226299300, −2.27759991576514638845229845020, −0.77721436915355036961649964790, 1.71644503586669524456608904952, 3.18301493576272530702818587033, 3.82964992408798899629724745625, 4.58064357418883232923225038575, 5.21091817893335988232345857490, 7.08839808945313282936833197098, 7.61606026907495915503868146519, 8.563809766257111304848499751199, 8.859163308784289487345969646462, 9.916721787452758379609082461661

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