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Properties

Label 2-1280-40.27-c1-0-16
Degree $2$
Conductor $1280$
Sign $0.973 - 0.229i$
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 − 2i)5-s + 3i·9-s + (1 + i)13-s + (3 + 3i)17-s + (−3 + 4i)25-s + 4·29-s + (7 − 7i)37-s + 8·41-s + (6 − 3i)45-s + 7i·49-s + (9 + 9i)53-s − 12i·61-s + (1 − 3i)65-s + (11 − 11i)73-s − 9·81-s + ⋯
L(s)  = 1  + (−0.447 − 0.894i)5-s + i·9-s + (0.277 + 0.277i)13-s + (0.727 + 0.727i)17-s + (−0.600 + 0.800i)25-s + 0.742·29-s + (1.15 − 1.15i)37-s + 1.24·41-s + (0.894 − 0.447i)45-s + i·49-s + (1.23 + 1.23i)53-s − 1.53i·61-s + (0.124 − 0.372i)65-s + (1.28 − 1.28i)73-s − 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.973 - 0.229i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 0.973 - 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.516198183\)
\(L(\frac12)\) \(\approx\) \(1.516198183\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 2i)T \)
good3 \( 1 - 3iT^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-1 - i)T + 13iT^{2} \)
17 \( 1 + (-3 - 3i)T + 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (-7 + 7i)T - 37iT^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (-9 - 9i)T + 53iT^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 12iT - 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-11 + 11i)T - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 - 16iT - 89T^{2} \)
97 \( 1 + (-13 - 13i)T + 97iT^{2} \)
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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

−9.568240642331605961562944213372, −8.868856216771511160194219694798, −7.938343535944210077954830740999, −7.62047579371549581888179827572, −6.26431745713657614381074924810, −5.42280719842764906931291849703, −4.55994791326550828512608548662, −3.77087942110642421376506493881, −2.35932758148223026856062009519, −1.07560883208194430273964421974, 0.820320506913134344902456481871, 2.63468761777757803659109401171, 3.41917159381795126682362195226, 4.31530874717417801948092227188, 5.58255744935582529467512340996, 6.43879918036070445624435376550, 7.12374153116788240812881369957, 7.955683167132138330383180203104, 8.800093670867458467264138813393, 9.825958827483972887153904020253

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