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Properties

Label 2-1280-20.7-c1-0-10
Degree $2$
Conductor $1280$
Sign $0.725 - 0.688i$
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  + (0.456 + 0.456i)3-s + (−0.456 − 2.18i)5-s + (−2.79 + 2.79i)7-s − 2.58i·9-s + 4.37i·11-s + (1.73 − 1.73i)13-s + (0.791 − 1.20i)15-s + (3 + 3i)17-s + 3.46·19-s − 2.55·21-s + (−0.791 − 0.791i)23-s + (−4.58 + 1.99i)25-s + (2.55 − 2.55i)27-s + 5.29i·29-s − 1.58i·31-s + ⋯
L(s)  = 1  + (0.263 + 0.263i)3-s + (−0.204 − 0.978i)5-s + (−1.05 + 1.05i)7-s − 0.860i·9-s + 1.31i·11-s + (0.480 − 0.480i)13-s + (0.204 − 0.312i)15-s + (0.727 + 0.727i)17-s + 0.794·19-s − 0.556·21-s + (−0.164 − 0.164i)23-s + (−0.916 + 0.399i)25-s + (0.490 − 0.490i)27-s + 0.982i·29-s − 0.284i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 - 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.725 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.725 - 0.688i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 0.725 - 0.688i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.501364660\)
\(L(\frac12)\) \(\approx\) \(1.501364660\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.456 + 2.18i)T \)
good3 \( 1 + (-0.456 - 0.456i)T + 3iT^{2} \)
7 \( 1 + (2.79 - 2.79i)T - 7iT^{2} \)
11 \( 1 - 4.37iT - 11T^{2} \)
13 \( 1 + (-1.73 + 1.73i)T - 13iT^{2} \)
17 \( 1 + (-3 - 3i)T + 17iT^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 + (0.791 + 0.791i)T + 23iT^{2} \)
29 \( 1 - 5.29iT - 29T^{2} \)
31 \( 1 + 1.58iT - 31T^{2} \)
37 \( 1 + (-5.19 - 5.19i)T + 37iT^{2} \)
41 \( 1 - 7.58T + 41T^{2} \)
43 \( 1 + (-8.29 - 8.29i)T + 43iT^{2} \)
47 \( 1 + (0.791 - 0.791i)T - 47iT^{2} \)
53 \( 1 + (2.64 - 2.64i)T - 53iT^{2} \)
59 \( 1 - 5.29T + 59T^{2} \)
61 \( 1 - 9.66T + 61T^{2} \)
67 \( 1 + (8.29 - 8.29i)T - 67iT^{2} \)
71 \( 1 + 13.5iT - 71T^{2} \)
73 \( 1 + (0.582 - 0.582i)T - 73iT^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + (-4.83 - 4.83i)T + 83iT^{2} \)
89 \( 1 - 3.16iT - 89T^{2} \)
97 \( 1 + (-0.582 - 0.582i)T + 97iT^{2} \)
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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

−9.538709097077781112582908853377, −9.172573579041375847365868525325, −8.305413624338686801290766864909, −7.43522998558471874479336667310, −6.24671556664992190346179455972, −5.67550668595993129976065843496, −4.58326970546059954857469065128, −3.64973315563314890300865915358, −2.71555211052575379549109709256, −1.16776725484021467645366660162, 0.72526471791695536047133783061, 2.52538297335098920023651730146, 3.36559306393131825937531644653, 4.06976475420929199685616595702, 5.61096942217865383263306897237, 6.32852680514097438500277191137, 7.36381746732754216538989878351, 7.58599935973365442101494145617, 8.726047620206968675447132097151, 9.734109758820410806960692146911

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