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Properties

Label 2-1280-8.5-c1-0-8
Degree $2$
Conductor $1280$
Sign $-0.707 - 0.707i$
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  + 2.82i·3-s − i·5-s + 2.82·7-s − 5.00·9-s + 5.65i·11-s − 2i·13-s + 2.82·15-s + 2·17-s + 8.00i·21-s − 2.82·23-s − 25-s − 5.65i·27-s + 6i·29-s + 5.65·31-s − 16.0·33-s + ⋯
L(s)  = 1  + 1.63i·3-s − 0.447i·5-s + 1.06·7-s − 1.66·9-s + 1.70i·11-s − 0.554i·13-s + 0.730·15-s + 0.485·17-s + 1.74i·21-s − 0.589·23-s − 0.200·25-s − 1.08i·27-s + 1.11i·29-s + 1.01·31-s − 2.78·33-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.658452185\)
\(L(\frac12)\) \(\approx\) \(1.658452185\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good3 \( 1 - 2.82iT - 3T^{2} \)
7 \( 1 - 2.82T + 7T^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 8.48iT - 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 11.3iT - 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 + 2.82iT - 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 - 2.82iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−9.880634095585036209324391514237, −9.482968859179606657162660724202, −8.338026772104313295440473409791, −7.907481727131496427315164156379, −6.60320709852443554873695040843, −5.21642104113449935245563080241, −4.88981382968949787566903416861, −4.22352158976485560969083953699, −3.09633989284459406934619731709, −1.65305633989897442053963579908, 0.71979849040965480667717737248, 1.84917373363336474364803851102, 2.79979616547071034496730591237, 4.05980074880724542032358770266, 5.55378169370215349425242754569, 6.08849272471504480749388027829, 6.96404900304427192139928943566, 7.81601374186909129794886578270, 8.235292219874550431507890998165, 9.034204763433559595355599062052

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