Properties

Degree $2$
Conductor $1024$
Sign $1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.07·3-s + 11.6·5-s + 2.67·7-s − 25.8·9-s + 63.9·11-s + 50.0·13-s − 12.4·15-s + 72.4·17-s + 27.4·19-s − 2.85·21-s − 139.·23-s + 10.3·25-s + 56.5·27-s − 93.3·29-s + 188.·31-s − 68.4·33-s + 31.0·35-s − 118.·37-s − 53.5·39-s + 104.·41-s + 44.5·43-s − 300.·45-s + 488.·47-s − 335.·49-s − 77.5·51-s − 211.·53-s + 743.·55-s + ⋯
L(s)  = 1  − 0.205·3-s + 1.04·5-s + 0.144·7-s − 0.957·9-s + 1.75·11-s + 1.06·13-s − 0.214·15-s + 1.03·17-s + 0.332·19-s − 0.0297·21-s − 1.26·23-s + 0.0826·25-s + 0.403·27-s − 0.598·29-s + 1.09·31-s − 0.361·33-s + 0.150·35-s − 0.528·37-s − 0.219·39-s + 0.398·41-s + 0.157·43-s − 0.996·45-s + 1.51·47-s − 0.979·49-s − 0.213·51-s − 0.548·53-s + 1.82·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $1$
Motivic weight: \(3\)
Character: $\chi_{1024} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.804765707\)
\(L(\frac12)\) \(\approx\) \(2.804765707\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 1.07T + 27T^{2} \)
5 \( 1 - 11.6T + 125T^{2} \)
7 \( 1 - 2.67T + 343T^{2} \)
11 \( 1 - 63.9T + 1.33e3T^{2} \)
13 \( 1 - 50.0T + 2.19e3T^{2} \)
17 \( 1 - 72.4T + 4.91e3T^{2} \)
19 \( 1 - 27.4T + 6.85e3T^{2} \)
23 \( 1 + 139.T + 1.21e4T^{2} \)
29 \( 1 + 93.3T + 2.43e4T^{2} \)
31 \( 1 - 188.T + 2.97e4T^{2} \)
37 \( 1 + 118.T + 5.06e4T^{2} \)
41 \( 1 - 104.T + 6.89e4T^{2} \)
43 \( 1 - 44.5T + 7.95e4T^{2} \)
47 \( 1 - 488.T + 1.03e5T^{2} \)
53 \( 1 + 211.T + 1.48e5T^{2} \)
59 \( 1 - 402.T + 2.05e5T^{2} \)
61 \( 1 - 322.T + 2.26e5T^{2} \)
67 \( 1 + 196.T + 3.00e5T^{2} \)
71 \( 1 - 453.T + 3.57e5T^{2} \)
73 \( 1 + 259.T + 3.89e5T^{2} \)
79 \( 1 + 323.T + 4.93e5T^{2} \)
83 \( 1 + 797.T + 5.71e5T^{2} \)
89 \( 1 + 866.T + 7.04e5T^{2} \)
97 \( 1 + 936.T + 9.12e5T^{2} \)
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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

−9.566449358293253371526906330906, −8.826819928214453141513089501983, −8.067989637357565257535457190789, −6.79124003860099910583007562232, −5.93751272540460068504086959115, −5.66432938630262054469243703483, −4.19651908209418389351757072040, −3.26744522557501404478961796726, −1.90377093689440825680717598967, −0.962157248111783288151881205879, 0.962157248111783288151881205879, 1.90377093689440825680717598967, 3.26744522557501404478961796726, 4.19651908209418389351757072040, 5.66432938630262054469243703483, 5.93751272540460068504086959115, 6.79124003860099910583007562232, 8.067989637357565257535457190789, 8.826819928214453141513089501983, 9.566449358293253371526906330906

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