Properties

Label 2-1280-1.1-c1-0-20
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  + 3.16·3-s − 5-s + 3.16·7-s + 7.00·9-s − 6·13-s − 3.16·15-s − 2·17-s + 6.32·19-s + 10.0·21-s + 3.16·23-s + 25-s + 12.6·27-s + 4·29-s + 6.32·31-s − 3.16·35-s − 2·37-s − 18.9·39-s − 3.16·43-s − 7.00·45-s − 9.48·47-s + 3.00·49-s − 6.32·51-s − 6·53-s + 20.0·57-s − 6.32·59-s − 2·61-s + 22.1·63-s + ⋯
L(s)  = 1  + 1.82·3-s − 0.447·5-s + 1.19·7-s + 2.33·9-s − 1.66·13-s − 0.816·15-s − 0.485·17-s + 1.45·19-s + 2.18·21-s + 0.659·23-s + 0.200·25-s + 2.43·27-s + 0.742·29-s + 1.13·31-s − 0.534·35-s − 0.328·37-s − 3.03·39-s − 0.482·43-s − 1.04·45-s − 1.38·47-s + 0.428·49-s − 0.885·51-s − 0.824·53-s + 2.64·57-s − 0.823·59-s − 0.256·61-s + 2.78·63-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

\(L(1)\) \(\approx\) \(3.267733182\)
\(L(\frac12)\) \(\approx\) \(3.267733182\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.16T + 3T^{2} \)
7 \( 1 - 3.16T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 6.32T + 19T^{2} \)
23 \( 1 - 3.16T + 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 - 6.32T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 3.16T + 43T^{2} \)
47 \( 1 + 9.48T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 6.32T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 9.48T + 67T^{2} \)
71 \( 1 - 6.32T + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 + 3.16T + 83T^{2} \)
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

−9.566160174959817216423668238033, −8.721249657251882844686648243933, −8.019184288594452912756046717291, −7.56368984657157567406817199118, −6.83035964707492582337739090398, −4.93670135674997377284350863438, −4.58073876691353591712906041194, −3.28516378942071903774413828489, −2.57085970041309012426694135381, −1.46548890812263135956192567274, 1.46548890812263135956192567274, 2.57085970041309012426694135381, 3.28516378942071903774413828489, 4.58073876691353591712906041194, 4.93670135674997377284350863438, 6.83035964707492582337739090398, 7.56368984657157567406817199118, 8.019184288594452912756046717291, 8.721249657251882844686648243933, 9.566160174959817216423668238033

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