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Properties

Label 2-1280-20.7-c1-0-41
Degree $2$
Conductor $1280$
Sign $-0.850 - 0.525i$
Analytic cond. $10.2208$
Root an. cond. $3.19700$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)3-s + (2 − i)5-s + (−1 + i)7-s − i·9-s + 4i·11-s + (−3 + 3i)13-s + (−3 − i)15-s + (−3 − 3i)17-s − 6·19-s + 2·21-s + (−3 − 3i)23-s + (3 − 4i)25-s + (−4 + 4i)27-s − 2i·29-s + 6i·31-s + ⋯
L(s)  = 1  + (−0.577 − 0.577i)3-s + (0.894 − 0.447i)5-s + (−0.377 + 0.377i)7-s − 0.333i·9-s + 1.20i·11-s + (−0.832 + 0.832i)13-s + (−0.774 − 0.258i)15-s + (−0.727 − 0.727i)17-s − 1.37·19-s + 0.436·21-s + (−0.625 − 0.625i)23-s + (0.600 − 0.800i)25-s + (−0.769 + 0.769i)27-s − 0.371i·29-s + 1.07i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.850 - 0.525i$
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: \(1\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ -0.850 - 0.525i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - i)T - 7iT^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 + (3 - 3i)T - 13iT^{2} \)
17 \( 1 + (3 + 3i)T + 17iT^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + (3 + 3i)T + 23iT^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 - 6iT - 31T^{2} \)
37 \( 1 + (3 + 3i)T + 37iT^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + (3 + 3i)T + 43iT^{2} \)
47 \( 1 + (-9 + 9i)T - 47iT^{2} \)
53 \( 1 + (5 - 5i)T - 53iT^{2} \)
59 \( 1 + 10T + 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 + (9 - 9i)T - 67iT^{2} \)
71 \( 1 + 6iT - 71T^{2} \)
73 \( 1 + (5 - 5i)T - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-3 - 3i)T + 83iT^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (7 + 7i)T + 97iT^{2} \)
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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

−9.199410844352709356249726533898, −8.632318355985944144310071390822, −7.14168588225264393809283129143, −6.74614447926115849110520562244, −5.98330834278771211460823296822, −4.99151588020140837812434783222, −4.23933687504990012576776971278, −2.46298842795261715058988739993, −1.76459039528730533636464057857, 0, 1.98380104940191069438811831423, 3.10793775980364593750376803825, 4.20491201335763035945327198834, 5.24034670808618061461712700234, 5.97872457856260773200563781721, 6.55790017168689678450898063968, 7.74162940427514495117161920140, 8.573553673398465119919932228310, 9.606299836933068250284171331592

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