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Properties

Label 2-1280-40.3-c1-0-9
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 + i)3-s + (−2 − i)5-s + (3 + 3i)7-s − i·9-s + 2·11-s + (−3 + 3i)13-s + (−1 − 3i)15-s + (1 − i)17-s + 4i·19-s + 6i·21-s + (1 − i)23-s + (3 + 4i)25-s + (4 − 4i)27-s + 10i·31-s + (2 + 2i)33-s + ⋯
L(s)  = 1  + (0.577 + 0.577i)3-s + (−0.894 − 0.447i)5-s + (1.13 + 1.13i)7-s − 0.333i·9-s + 0.603·11-s + (−0.832 + 0.832i)13-s + (−0.258 − 0.774i)15-s + (0.242 − 0.242i)17-s + 0.917i·19-s + 1.30i·21-s + (0.208 − 0.208i)23-s + (0.600 + 0.800i)25-s + (0.769 − 0.769i)27-s + 1.79i·31-s + (0.348 + 0.348i)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: $\chi_{1280} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 0.229 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.862729411\)
\(L(\frac12)\) \(\approx\) \(1.862729411\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - i)T + 3iT^{2} \)
7 \( 1 + (-3 - 3i)T + 7iT^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (3 - 3i)T - 13iT^{2} \)
17 \( 1 + (-1 + i)T - 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (-1 + i)T - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 10iT - 31T^{2} \)
37 \( 1 + (-1 - i)T + 37iT^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + (5 + 5i)T + 43iT^{2} \)
47 \( 1 + (3 + 3i)T + 47iT^{2} \)
53 \( 1 + (5 - 5i)T - 53iT^{2} \)
59 \( 1 - 12iT - 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 + (-1 + i)T - 67iT^{2} \)
71 \( 1 + 2iT - 71T^{2} \)
73 \( 1 + (1 + i)T + 73iT^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (-5 - 5i)T + 83iT^{2} \)
89 \( 1 - 16iT - 89T^{2} \)
97 \( 1 + (3 - 3i)T - 97iT^{2} \)
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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

−9.517903836074491164888341976728, −8.961028642058998952491036516737, −8.434780925515295408288556011918, −7.63928529242605102454852811328, −6.62253915419771897206932509035, −5.37051519172201753730823556759, −4.62815617944922678664482831841, −3.87790605644655920958436775839, −2.78168196671581252471151035123, −1.46952488541854550988251911405, 0.797254719091624168491719672082, 2.16445420791154516633389851178, 3.26897770244475509297242269435, 4.32206740114971620348748626768, 5.00160488029321986850256738771, 6.48760004088276344169223511844, 7.47452493913816062160691201612, 7.69993112328264032091837544466, 8.286360693471048380898984215120, 9.464062785573687454981868709775

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