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

\(L(2)\) \(\approx\) \(0.7136084607\)
\(L(\frac12)\) \(\approx\) \(0.7136084607\)
\(L(\frac{5}{2})\) 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

−9.672739000274910270600223123991, −8.371313042719906757132948321525, −7.953847432262216838344651971527, −7.43190024346947859269384380310, −6.03101046301288799923408537666, −5.19444674393403632060217116116, −4.28827815129986289360720715341, −3.08736760774475990019420857726, −2.37911539515792419699052942342, −0.41400873766664121318632781215, 0.41400873766664121318632781215, 2.37911539515792419699052942342, 3.08736760774475990019420857726, 4.28827815129986289360720715341, 5.19444674393403632060217116116, 6.03101046301288799923408537666, 7.43190024346947859269384380310, 7.953847432262216838344651971527, 8.371313042719906757132948321525, 9.672739000274910270600223123991

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