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Properties

Label 2-1280-40.29-c1-0-23
Degree $2$
Conductor $1280$
Sign $0.856 - 0.515i$
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  + 1.70·3-s + (0.539 − 2.17i)5-s + 2.63i·7-s − 0.0783·9-s + 5.41i·11-s + 6.34·13-s + (0.921 − 3.70i)15-s + 3.41i·17-s + 3.26i·19-s + 4.49i·21-s − 1.36i·23-s + (−4.41 − 2.34i)25-s − 5.26·27-s − 2i·29-s + 4.68·31-s + ⋯
L(s)  = 1  + 0.986·3-s + (0.241 − 0.970i)5-s + 0.994i·7-s − 0.0261·9-s + 1.63i·11-s + 1.75·13-s + (0.237 − 0.957i)15-s + 0.829i·17-s + 0.748i·19-s + 0.981i·21-s − 0.285i·23-s + (−0.883 − 0.468i)25-s − 1.01·27-s − 0.371i·29-s + 0.840·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.856 - 0.515i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 0.856 - 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.448805061\)
\(L(\frac12)\) \(\approx\) \(2.448805061\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.539 + 2.17i)T \)
good3 \( 1 - 1.70T + 3T^{2} \)
7 \( 1 - 2.63iT - 7T^{2} \)
11 \( 1 - 5.41iT - 11T^{2} \)
13 \( 1 - 6.34T + 13T^{2} \)
17 \( 1 - 3.41iT - 17T^{2} \)
19 \( 1 - 3.26iT - 19T^{2} \)
23 \( 1 + 1.36iT - 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 - 4.68T + 31T^{2} \)
37 \( 1 + 5.75T + 37T^{2} \)
41 \( 1 - 7.75T + 41T^{2} \)
43 \( 1 - 4.44T + 43T^{2} \)
47 \( 1 + 4.78iT - 47T^{2} \)
53 \( 1 + 1.65T + 53T^{2} \)
59 \( 1 - 3.26iT - 59T^{2} \)
61 \( 1 + 2.49iT - 61T^{2} \)
67 \( 1 - 7.86T + 67T^{2} \)
71 \( 1 + 6.15T + 71T^{2} \)
73 \( 1 + 13.5iT - 73T^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 + 14.9T + 83T^{2} \)
89 \( 1 + 8.52T + 89T^{2} \)
97 \( 1 + 4.58iT - 97T^{2} \)
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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

−9.516768034929246182854735691707, −8.728906890519987449682233534604, −8.454757129114537129131160554262, −7.60590078738788651941234883667, −6.23300564415862171317710722028, −5.64179141399862291234752299759, −4.47388609697318311892120893156, −3.65647529477294157265530110032, −2.33184866609757018041268796582, −1.57525610902437685051853742601, 0.998787871345519049870340245082, 2.66505672829299754297055220035, 3.33981540516744409999881298230, 3.99017679348804031028670187841, 5.60532697126220148590796704965, 6.36271506798405530350788427430, 7.21278864765205398535370438048, 8.073152264212579658932248489576, 8.742080818906847600076459821911, 9.448981773807819261551257181556

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