Properties

Label 2-1280-40.29-c1-0-10
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  + 1.23·3-s + 2.23i·5-s + 5.23i·7-s − 1.47·9-s + 2.76i·15-s + 6.47i·21-s − 7.70i·23-s − 5.00·25-s − 5.52·27-s + 6i·29-s − 11.7·35-s − 4.47·41-s + 6.76·43-s − 3.29i·45-s − 0.291i·47-s + ⋯
L(s)  = 1  + 0.713·3-s + 0.999i·5-s + 1.97i·7-s − 0.490·9-s + 0.713i·15-s + 1.41i·21-s − 1.60i·23-s − 1.00·25-s − 1.06·27-s + 1.11i·29-s − 1.97·35-s − 0.698·41-s + 1.03·43-s − 0.490i·45-s − 0.0425i·47-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: $\chi_{1280} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.576390394\)
\(L(\frac12)\) \(\approx\) \(1.576390394\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23iT \)
good3 \( 1 - 1.23T + 3T^{2} \)
7 \( 1 - 5.23iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 7.70iT - 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 4.47T + 41T^{2} \)
43 \( 1 - 6.76T + 43T^{2} \)
47 \( 1 + 0.291iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 13.4iT - 61T^{2} \)
67 \( 1 - 14.1T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 4.29T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−9.840768337423665582616435790770, −8.893268082341229214547202468850, −8.615168468455555011805324002991, −7.67773800126919580419352742776, −6.56501129617566401945718576998, −5.91632018248476031578471745370, −5.03345452595656004052874214796, −3.55037195662370158802562300755, −2.66543913310271005286843022162, −2.22142644494806211316697372181, 0.57622455115973410802870284720, 1.80807590031542927953216856909, 3.39572482943623920417532168379, 4.02388153881232411507310705479, 4.94917232199680345463683816598, 6.00411614949344167019037991834, 7.19766948759104154443544285016, 7.83105097280376366455529591520, 8.415766000425388784991716352467, 9.508483508625329851100272417971

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