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Properties

Label 2-1280-80.43-c1-0-10
Degree $2$
Conductor $1280$
Sign $0.969 - 0.245i$
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  − 2.41i·3-s + (−2.21 + 0.328i)5-s + (−0.988 + 0.988i)7-s − 2.85·9-s + (−3.82 + 3.82i)11-s − 1.72·13-s + (0.794 + 5.35i)15-s + (5.54 − 5.54i)17-s + (−0.392 + 0.392i)19-s + (2.39 + 2.39i)21-s + (4.28 + 4.28i)23-s + (4.78 − 1.45i)25-s − 0.353i·27-s + (4.24 + 4.24i)29-s + 6.43i·31-s + ⋯
L(s)  = 1  − 1.39i·3-s + (−0.989 + 0.146i)5-s + (−0.373 + 0.373i)7-s − 0.951·9-s + (−1.15 + 1.15i)11-s − 0.478·13-s + (0.205 + 1.38i)15-s + (1.34 − 1.34i)17-s + (−0.0900 + 0.0900i)19-s + (0.521 + 0.521i)21-s + (0.893 + 0.893i)23-s + (0.956 − 0.290i)25-s − 0.0681i·27-s + (0.787 + 0.787i)29-s + 1.15i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.969 - 0.245i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 0.969 - 0.245i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9370125877\)
\(L(\frac12)\) \(\approx\) \(0.9370125877\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 - 0.328i)T \)
good3 \( 1 + 2.41iT - 3T^{2} \)
7 \( 1 + (0.988 - 0.988i)T - 7iT^{2} \)
11 \( 1 + (3.82 - 3.82i)T - 11iT^{2} \)
13 \( 1 + 1.72T + 13T^{2} \)
17 \( 1 + (-5.54 + 5.54i)T - 17iT^{2} \)
19 \( 1 + (0.392 - 0.392i)T - 19iT^{2} \)
23 \( 1 + (-4.28 - 4.28i)T + 23iT^{2} \)
29 \( 1 + (-4.24 - 4.24i)T + 29iT^{2} \)
31 \( 1 - 6.43iT - 31T^{2} \)
37 \( 1 + 0.399T + 37T^{2} \)
41 \( 1 + 2.41iT - 41T^{2} \)
43 \( 1 - 1.73T + 43T^{2} \)
47 \( 1 + (2.36 + 2.36i)T + 47iT^{2} \)
53 \( 1 - 6.63iT - 53T^{2} \)
59 \( 1 + (-6.32 - 6.32i)T + 59iT^{2} \)
61 \( 1 + (-7.89 + 7.89i)T - 61iT^{2} \)
67 \( 1 - 0.547T + 67T^{2} \)
71 \( 1 + 2.49T + 71T^{2} \)
73 \( 1 + (5.30 - 5.30i)T - 73iT^{2} \)
79 \( 1 + 14.1T + 79T^{2} \)
83 \( 1 - 9.94iT - 83T^{2} \)
89 \( 1 - 8.69T + 89T^{2} \)
97 \( 1 + (3.81 - 3.81i)T - 97iT^{2} \)
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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

−9.708441094389511270697435561848, −8.655603695992780196631201708940, −7.72583869292908116943937820955, −7.29086131689879169840821938733, −6.85811414229697156947076778010, −5.47522491696850337641055892629, −4.78527619848123516733313511548, −3.20131179747702716077037299786, −2.51237226934070672222608386930, −1.04720585074573216453482954171, 0.48308018516500025283789561731, 2.90463892218425625265315765562, 3.59406288595544638602166550037, 4.39049747633401826212726291441, 5.21814812112265209668734876422, 6.10775402236734600588564862058, 7.39079503539570429798221260894, 8.228449341520348560305700459002, 8.708457306007850370773591693345, 9.992299272373617624475208240464

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