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Properties

Label 2-1280-16.5-c1-0-19
Degree $2$
Conductor $1280$
Sign $0.991 - 0.130i$
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  + (1.07 − 1.07i)3-s + (0.707 + 0.707i)5-s − 0.103i·7-s + 0.696i·9-s + (2.21 + 2.21i)11-s + (−0.931 + 0.931i)13-s + 1.51·15-s + 2.54·17-s + (2.94 − 2.94i)19-s + (−0.110 − 0.110i)21-s − 1.07i·23-s + 1.00i·25-s + (3.96 + 3.96i)27-s + (−6.32 + 6.32i)29-s + 0.635·31-s + ⋯
L(s)  = 1  + (0.619 − 0.619i)3-s + (0.316 + 0.316i)5-s − 0.0390i·7-s + 0.232i·9-s + (0.667 + 0.667i)11-s + (−0.258 + 0.258i)13-s + 0.391·15-s + 0.617·17-s + (0.675 − 0.675i)19-s + (−0.0242 − 0.0242i)21-s − 0.224i·23-s + 0.200i·25-s + (0.763 + 0.763i)27-s + (−1.17 + 1.17i)29-s + 0.114·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.991 - 0.130i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 0.991 - 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.284498390\)
\(L(\frac12)\) \(\approx\) \(2.284498390\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-1.07 + 1.07i)T - 3iT^{2} \)
7 \( 1 + 0.103iT - 7T^{2} \)
11 \( 1 + (-2.21 - 2.21i)T + 11iT^{2} \)
13 \( 1 + (0.931 - 0.931i)T - 13iT^{2} \)
17 \( 1 - 2.54T + 17T^{2} \)
19 \( 1 + (-2.94 + 2.94i)T - 19iT^{2} \)
23 \( 1 + 1.07iT - 23T^{2} \)
29 \( 1 + (6.32 - 6.32i)T - 29iT^{2} \)
31 \( 1 - 0.635T + 31T^{2} \)
37 \( 1 + (-3.76 - 3.76i)T + 37iT^{2} \)
41 \( 1 + 5.23iT - 41T^{2} \)
43 \( 1 + (-5.85 - 5.85i)T + 43iT^{2} \)
47 \( 1 - 9.86T + 47T^{2} \)
53 \( 1 + (6.04 + 6.04i)T + 53iT^{2} \)
59 \( 1 + (6.78 + 6.78i)T + 59iT^{2} \)
61 \( 1 + (5.58 - 5.58i)T - 61iT^{2} \)
67 \( 1 + (0.187 - 0.187i)T - 67iT^{2} \)
71 \( 1 + 2.42iT - 71T^{2} \)
73 \( 1 + 14.1iT - 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + (-1.03 + 1.03i)T - 83iT^{2} \)
89 \( 1 - 6.75iT - 89T^{2} \)
97 \( 1 + 17.9T + 97T^{2} \)
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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

−9.424057174002568928411495025493, −9.050768577517630868382605871220, −7.84547791482512870108457387510, −7.32266230659267720240456523727, −6.62724239400393785952636862064, −5.51735659792169156409184133363, −4.55675922752012691694900822149, −3.33134850764205909931743135809, −2.36665251100054955806143660098, −1.38401645359076981943934606845, 1.04102445622231585095570298447, 2.57578039712713293472573874158, 3.61135478252193320288031346814, 4.25564299393508500628793466211, 5.58176277009136734041252399875, 6.08276729969224527610237799830, 7.39863560910798171439375167786, 8.127947921702672511872045576893, 9.129558493796410809567055183800, 9.437246235754519932595059676670

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