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Properties

Label 2-1280-16.13-c1-0-12
Degree $2$
Conductor $1280$
Sign $0.923 + 0.382i$
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  + (−0.742 − 0.742i)3-s + (0.707 − 0.707i)5-s + 0.463i·7-s − 1.89i·9-s + (−2.04 + 2.04i)11-s + (3.94 + 3.94i)13-s − 1.04·15-s + 1.34·17-s + (4.94 + 4.94i)19-s + (0.344 − 0.344i)21-s − 3.84i·23-s − 1.00i·25-s + (−3.63 + 3.63i)27-s + (−1.09 − 1.09i)29-s + 3.17·31-s + ⋯
L(s)  = 1  + (−0.428 − 0.428i)3-s + (0.316 − 0.316i)5-s + 0.175i·7-s − 0.632i·9-s + (−0.617 + 0.617i)11-s + (1.09 + 1.09i)13-s − 0.270·15-s + 0.326·17-s + (1.13 + 1.13i)19-s + (0.0751 − 0.0751i)21-s − 0.802i·23-s − 0.200i·25-s + (−0.699 + 0.699i)27-s + (−0.204 − 0.204i)29-s + 0.569·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 0.923 + 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.562811502\)
\(L(\frac12)\) \(\approx\) \(1.562811502\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.742 + 0.742i)T + 3iT^{2} \)
7 \( 1 - 0.463iT - 7T^{2} \)
11 \( 1 + (2.04 - 2.04i)T - 11iT^{2} \)
13 \( 1 + (-3.94 - 3.94i)T + 13iT^{2} \)
17 \( 1 - 1.34T + 17T^{2} \)
19 \( 1 + (-4.94 - 4.94i)T + 19iT^{2} \)
23 \( 1 + 3.84iT - 23T^{2} \)
29 \( 1 + (1.09 + 1.09i)T + 29iT^{2} \)
31 \( 1 - 3.17T + 31T^{2} \)
37 \( 1 + (-6.89 + 6.89i)T - 37iT^{2} \)
41 \( 1 + 2.89iT - 41T^{2} \)
43 \( 1 + (1.62 - 1.62i)T - 43iT^{2} \)
47 \( 1 - 3.29T + 47T^{2} \)
53 \( 1 + (-3.29 + 3.29i)T - 53iT^{2} \)
59 \( 1 + (6.84 - 6.84i)T - 59iT^{2} \)
61 \( 1 + (-3.44 - 3.44i)T + 61iT^{2} \)
67 \( 1 + (5.93 + 5.93i)T + 67iT^{2} \)
71 \( 1 + 15.8iT - 71T^{2} \)
73 \( 1 - 0.556iT - 73T^{2} \)
79 \( 1 - 4.23T + 79T^{2} \)
83 \( 1 + (-7.06 - 7.06i)T + 83iT^{2} \)
89 \( 1 - 17.3iT - 89T^{2} \)
97 \( 1 - 9.04T + 97T^{2} \)
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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

−9.527264376503329321831392370603, −8.936706518950543141809363528321, −7.918141209388211680426500815469, −7.11705853561197617937403972256, −6.14729419619569975736242748043, −5.68755158597244803589372675003, −4.50456939580221467675243751363, −3.54042673733730780349540623335, −2.10900095698094288477912487394, −0.992652691564249763853431609052, 0.977331002802262902517450373174, 2.68999123583406756790339652529, 3.48935443377051880279365721143, 4.81386667703069006841397014245, 5.52785424035294472184514229647, 6.15130801594449633192454294056, 7.40553515981532005813583977646, 8.008212271264068880726513745622, 8.949760107815794390394221339798, 9.987898695419440737242716734268

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