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Properties

Label 2-1280-40.19-c0-0-1
Degree $2$
Conductor $1280$
Sign $0.707 + 0.707i$
Analytic cond. $0.638803$
Root an. cond. $0.799251$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·5-s + 9-s − 25-s − 2i·29-s + 2·41-s − i·45-s − 49-s + 2i·61-s + 81-s − 2·89-s + 2i·101-s + 2i·109-s + ⋯
L(s)  = 1  − i·5-s + 9-s − 25-s − 2i·29-s + 2·41-s − i·45-s − 49-s + 2i·61-s + 81-s − 2·89-s + 2i·101-s + 2i·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.131734578\)
\(L(\frac12)\) \(\approx\) \(1.131734578\)
\(L(1)\) 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 \( 1 + iT \)
good3 \( 1 - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + 2iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 2T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - 2iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.667459165055266968595816369154, −9.081758883693930174934303337483, −8.064006775761831723103436634520, −7.51041393700793237534338313363, −6.39194979146782392669591705024, −5.54359395567518135212106836468, −4.50307621547822240437210255771, −3.97423021975834343415153124354, −2.38980190813354691460753359718, −1.12985572546244192274549107844, 1.64330031213782616999717196204, 2.89733515607916253952647196850, 3.82502464749591129704786151542, 4.82352186882573125275769387637, 5.93330150882845923892834145932, 6.83982491707451663541364549824, 7.33608252330103242552435231391, 8.246088991136250399594171153945, 9.362415640710904540750174732075, 9.952380009061971476016893250330

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