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Properties

Label 2-1280-40.27-c1-0-39
Degree $2$
Conductor $1280$
Sign $0.229 + 0.973i$
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  + (1.73 − 1.73i)3-s + (2 − i)5-s + (1.73 − 1.73i)7-s − 2.99i·9-s + 3.46·11-s + (−1 − i)13-s + (1.73 − 5.19i)15-s + (1 + i)17-s + 6.92i·19-s − 5.99i·21-s + (−1.73 − 1.73i)23-s + (3 − 4i)25-s − 4·29-s + 3.46i·31-s + (5.99 − 5.99i)33-s + ⋯
L(s)  = 1  + (0.999 − 0.999i)3-s + (0.894 − 0.447i)5-s + (0.654 − 0.654i)7-s − 0.999i·9-s + 1.04·11-s + (−0.277 − 0.277i)13-s + (0.447 − 1.34i)15-s + (0.242 + 0.242i)17-s + 1.58i·19-s − 1.30i·21-s + (−0.361 − 0.361i)23-s + (0.600 − 0.800i)25-s − 0.742·29-s + 0.622i·31-s + (1.04 − 1.04i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 0.229 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.036096149\)
\(L(\frac12)\) \(\approx\) \(3.036096149\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + i)T \)
good3 \( 1 + (-1.73 + 1.73i)T - 3iT^{2} \)
7 \( 1 + (-1.73 + 1.73i)T - 7iT^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 + (1 + i)T + 13iT^{2} \)
17 \( 1 + (-1 - i)T + 17iT^{2} \)
19 \( 1 - 6.92iT - 19T^{2} \)
23 \( 1 + (1.73 + 1.73i)T + 23iT^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + (5 - 5i)T - 37iT^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + (1.73 - 1.73i)T - 43iT^{2} \)
47 \( 1 + (1.73 - 1.73i)T - 47iT^{2} \)
53 \( 1 + (7 + 7i)T + 53iT^{2} \)
59 \( 1 - 6.92iT - 59T^{2} \)
61 \( 1 + 6iT - 61T^{2} \)
67 \( 1 + (5.19 + 5.19i)T + 67iT^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (-7 + 7i)T - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (12.1 - 12.1i)T - 83iT^{2} \)
89 \( 1 - 8iT - 89T^{2} \)
97 \( 1 + (7 + 7i)T + 97iT^{2} \)
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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

−9.435012313204121092761444810891, −8.389767983189687045933286466531, −8.109927605313039924114717107155, −7.10036458298342385290836432137, −6.36958317014230045583027012553, −5.36384834673605009816578433895, −4.21059687140627401186053286543, −3.15169424762984126071456811197, −1.75791555210869630123897030211, −1.39638625675540828333485341571, 1.82968493364292697137694916842, 2.69345042220394502671205957342, 3.67030678293651480734874533900, 4.67693418926345662381358637912, 5.48973860372372011744900298779, 6.55066560635412489296890055329, 7.45793580869095546246580441300, 8.607182397100632813797407148349, 9.203860668932615252031428695384, 9.526625433395182113369164723701

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