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Properties

Label 2-1280-16.13-c1-0-17
Degree $2$
Conductor $1280$
Sign $0.608 - 0.793i$
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  + (2.07 + 2.07i)3-s + (−0.707 + 0.707i)5-s − 4.34i·7-s + 5.59i·9-s + (2.66 − 2.66i)11-s + (0.482 + 0.482i)13-s − 2.93·15-s + 0.353·17-s + (5.39 + 5.39i)19-s + (9.00 − 9.00i)21-s + 6.62i·23-s − 1.00i·25-s + (−5.38 + 5.38i)27-s + (3.42 + 3.42i)29-s + 0.635·31-s + ⋯
L(s)  = 1  + (1.19 + 1.19i)3-s + (−0.316 + 0.316i)5-s − 1.64i·7-s + 1.86i·9-s + (0.803 − 0.803i)11-s + (0.133 + 0.133i)13-s − 0.757·15-s + 0.0856·17-s + (1.23 + 1.23i)19-s + (1.96 − 1.96i)21-s + 1.38i·23-s − 0.200i·25-s + (−1.03 + 1.03i)27-s + (0.636 + 0.636i)29-s + 0.114·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.608 - 0.793i$
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: \((2,\ 1280,\ (\ :1/2),\ 0.608 - 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.559253452\)
\(L(\frac12)\) \(\approx\) \(2.559253452\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.07 - 2.07i)T + 3iT^{2} \)
7 \( 1 + 4.34iT - 7T^{2} \)
11 \( 1 + (-2.66 + 2.66i)T - 11iT^{2} \)
13 \( 1 + (-0.482 - 0.482i)T + 13iT^{2} \)
17 \( 1 - 0.353T + 17T^{2} \)
19 \( 1 + (-5.39 - 5.39i)T + 19iT^{2} \)
23 \( 1 - 6.62iT - 23T^{2} \)
29 \( 1 + (-3.42 - 3.42i)T + 29iT^{2} \)
31 \( 1 - 0.635T + 31T^{2} \)
37 \( 1 + (-3.13 + 3.13i)T - 37iT^{2} \)
41 \( 1 + 2.33iT - 41T^{2} \)
43 \( 1 + (7.59 - 7.59i)T - 43iT^{2} \)
47 \( 1 - 5.41T + 47T^{2} \)
53 \( 1 + (-4.49 + 4.49i)T - 53iT^{2} \)
59 \( 1 + (-9.46 + 9.46i)T - 59iT^{2} \)
61 \( 1 + (6.21 + 6.21i)T + 61iT^{2} \)
67 \( 1 + (-0.362 - 0.362i)T + 67iT^{2} \)
71 \( 1 - 7.32iT - 71T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 + 8.13T + 79T^{2} \)
83 \( 1 + (8.41 + 8.41i)T + 83iT^{2} \)
89 \( 1 - 4.55iT - 89T^{2} \)
97 \( 1 - 2.17T + 97T^{2} \)
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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

−9.827366560862541572646761487168, −9.118707695921985401849490279935, −8.122232486794528619005581473169, −7.63360170109186943899770712388, −6.68205231175584392906256136641, −5.36072128987111090108375084266, −4.21019859327833133927643157878, −3.63629316641836377890607459470, −3.18637472797123083286467818520, −1.33844734552857222677633286735, 1.16528915467203388910260674870, 2.39815508566281578232173196068, 2.89151002497264776349168069305, 4.28433555022284616784744168456, 5.43491346191568238435010806363, 6.52096155656130225551789522400, 7.14193735693355369761596809373, 8.068821623091770907810628111994, 8.789402200848939358072436207383, 9.076758580987307231213940460527

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