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Properties

Label 2-1280-8.5-c3-0-27
Degree $2$
Conductor $1280$
Sign $-0.707 - 0.707i$
Analytic cond. $75.5224$
Root an. cond. $8.69036$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.47i·3-s + 5i·5-s + 31.3·7-s + 6.99·9-s + 8.94i·11-s − 62i·13-s − 22.3·15-s − 46·17-s + 107. i·19-s + 140i·21-s − 192.·23-s − 25·25-s + 152. i·27-s − 90i·29-s − 152.·31-s + ⋯
L(s)  = 1  + 0.860i·3-s + 0.447i·5-s + 1.69·7-s + 0.259·9-s + 0.245i·11-s − 1.32i·13-s − 0.384·15-s − 0.656·17-s + 1.29i·19-s + 1.45i·21-s − 1.74·23-s − 0.200·25-s + 1.08i·27-s − 0.576i·29-s − 0.880·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(75.5224\)
Root analytic conductor: \(8.69036\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :3/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.155891258\)
\(L(\frac12)\) \(\approx\) \(2.155891258\)
\(L(\frac{5}{2})\) 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 \( 1 - 5iT \)
good3 \( 1 - 4.47iT - 27T^{2} \)
7 \( 1 - 31.3T + 343T^{2} \)
11 \( 1 - 8.94iT - 1.33e3T^{2} \)
13 \( 1 + 62iT - 2.19e3T^{2} \)
17 \( 1 + 46T + 4.91e3T^{2} \)
19 \( 1 - 107. iT - 6.85e3T^{2} \)
23 \( 1 + 192.T + 1.21e4T^{2} \)
29 \( 1 + 90iT - 2.43e4T^{2} \)
31 \( 1 + 152.T + 2.97e4T^{2} \)
37 \( 1 - 214iT - 5.06e4T^{2} \)
41 \( 1 - 10T + 6.89e4T^{2} \)
43 \( 1 - 67.0iT - 7.95e4T^{2} \)
47 \( 1 - 398.T + 1.03e5T^{2} \)
53 \( 1 - 678iT - 1.48e5T^{2} \)
59 \( 1 - 411. iT - 2.05e5T^{2} \)
61 \( 1 - 250iT - 2.26e5T^{2} \)
67 \( 1 - 49.1iT - 3.00e5T^{2} \)
71 \( 1 - 366.T + 3.57e5T^{2} \)
73 \( 1 + 522T + 3.89e5T^{2} \)
79 \( 1 - 876.T + 4.93e5T^{2} \)
83 \( 1 - 380. iT - 5.71e5T^{2} \)
89 \( 1 + 970T + 7.04e5T^{2} \)
97 \( 1 + 934T + 9.12e5T^{2} \)
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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

−9.836298123709377668574464757430, −8.739457140757484667286748448736, −7.891922882013459585771966553157, −7.49490885249302586412625899506, −6.04600888502554303506236424203, −5.31999676510117147675401889573, −4.37893453975748396549307363199, −3.79535420516868782820266454424, −2.38715769650539617571227666553, −1.34829455917969117263002784972, 0.48551015029238458257720659547, 1.79226970452170293497860322022, 2.05826005077091904047821762205, 4.05262421936386100360940474467, 4.64940045137544488462725649025, 5.60618084651259818025515095374, 6.73554672920332915683664715527, 7.33950219276853145308792160334, 8.186822949506990530484028470179, 8.780416097929618146626344343975

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