Properties

Label 2-1280-1.1-c1-0-13
Degree $2$
Conductor $1280$
Sign $1$
Analytic cond. $10.2208$
Root an. cond. $3.19700$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·3-s − 5-s + 4.24·7-s − 0.999·9-s + 2.82·11-s + 6·13-s − 1.41·15-s − 6·17-s + 6·21-s + 4.24·23-s + 25-s − 5.65·27-s − 8.48·31-s + 4.00·33-s − 4.24·35-s + 6·37-s + 8.48·39-s + 4.24·43-s + 0.999·45-s + 4.24·47-s + 10.9·49-s − 8.48·51-s + 6·53-s − 2.82·55-s − 11.3·59-s + 6·61-s − 4.24·63-s + ⋯
L(s)  = 1  + 0.816·3-s − 0.447·5-s + 1.60·7-s − 0.333·9-s + 0.852·11-s + 1.66·13-s − 0.365·15-s − 1.45·17-s + 1.30·21-s + 0.884·23-s + 0.200·25-s − 1.08·27-s − 1.52·31-s + 0.696·33-s − 0.717·35-s + 0.986·37-s + 1.35·39-s + 0.646·43-s + 0.149·45-s + 0.618·47-s + 1.57·49-s − 1.18·51-s + 0.824·53-s − 0.381·55-s − 1.47·59-s + 0.768·61-s − 0.534·63-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.513723464\)
\(L(\frac12)\) \(\approx\) \(2.513723464\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good3 \( 1 - 1.41T + 3T^{2} \)
7 \( 1 - 4.24T + 7T^{2} \)
11 \( 1 - 2.82T + 11T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 4.24T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 8.48T + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 4.24T + 43T^{2} \)
47 \( 1 - 4.24T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 + 8.48T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−9.135738007135489841500532466931, −8.872462334402988355068017525881, −8.206639157502084528486600815541, −7.45821065340662889937664742530, −6.43270933122789841211637865148, −5.39347463419222355222088833077, −4.28205376380185187932448349488, −3.68457874219985765989446656655, −2.35306015800440012562474726032, −1.28485458580888682346600691311, 1.28485458580888682346600691311, 2.35306015800440012562474726032, 3.68457874219985765989446656655, 4.28205376380185187932448349488, 5.39347463419222355222088833077, 6.43270933122789841211637865148, 7.45821065340662889937664742530, 8.206639157502084528486600815541, 8.872462334402988355068017525881, 9.135738007135489841500532466931

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