Properties

Label 2-1280-40.29-c1-0-24
Degree $2$
Conductor $1280$
Sign $-0.601 + 0.798i$
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  − 2.90·3-s + (−2.21 + 0.311i)5-s − 3.52i·7-s + 5.42·9-s + 3.80i·11-s + 2.62·13-s + (6.42 − 0.903i)15-s + 5.80i·17-s − 5.05i·19-s + 10.2i·21-s + 0.474i·23-s + (4.80 − 1.37i)25-s − 7.05·27-s + 2i·29-s − 2.75·31-s + ⋯
L(s)  = 1  − 1.67·3-s + (−0.990 + 0.139i)5-s − 1.33i·7-s + 1.80·9-s + 1.14i·11-s + 0.727·13-s + (1.65 − 0.233i)15-s + 1.40i·17-s − 1.15i·19-s + 2.23i·21-s + 0.0989i·23-s + (0.961 − 0.275i)25-s − 1.35·27-s + 0.371i·29-s − 0.494·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.601 + 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.601 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.601 + 0.798i$
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: \((2,\ 1280,\ (\ :1/2),\ -0.601 + 0.798i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3312598261\)
\(L(\frac12)\) \(\approx\) \(0.3312598261\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 - 0.311i)T \)
good3 \( 1 + 2.90T + 3T^{2} \)
7 \( 1 + 3.52iT - 7T^{2} \)
11 \( 1 - 3.80iT - 11T^{2} \)
13 \( 1 - 2.62T + 13T^{2} \)
17 \( 1 - 5.80iT - 17T^{2} \)
19 \( 1 + 5.05iT - 19T^{2} \)
23 \( 1 - 0.474iT - 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 + 2.75T + 31T^{2} \)
37 \( 1 - 7.18T + 37T^{2} \)
41 \( 1 + 5.18T + 41T^{2} \)
43 \( 1 + 1.95T + 43T^{2} \)
47 \( 1 + 5.33iT - 47T^{2} \)
53 \( 1 + 5.37T + 53T^{2} \)
59 \( 1 + 5.05iT - 59T^{2} \)
61 \( 1 + 12.2iT - 61T^{2} \)
67 \( 1 + 7.76T + 67T^{2} \)
71 \( 1 - 4.85T + 71T^{2} \)
73 \( 1 + 6.66iT - 73T^{2} \)
79 \( 1 - 5.24T + 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−9.737741712956297337712430384872, −8.387381808916097415034534500188, −7.42317380216201800846709230739, −6.88761371469323909821033198649, −6.22285847895742343930524631207, −4.96275293020163801368425196248, −4.36704722236192362734842318888, −3.60998479776990115329557161954, −1.45294914689524267233933843123, −0.22594354613682051967255643017, 1.07546422311426779508781069030, 2.94210692351814796099076475427, 4.11055146910283247759856693277, 5.13077341445910684473441202879, 5.77915876504415876508341224264, 6.34587126817793682194049259692, 7.43119095880157122588968896722, 8.344293895602701374575988251743, 9.087862452015664506237653150114, 10.14232963033502136130391053739

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