Properties

Label 2-2e6-1.1-c9-0-10
Degree $2$
Conductor $64$
Sign $1$
Analytic cond. $32.9622$
Root an. cond. $5.74127$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 228·3-s + 666·5-s + 6.32e3·7-s + 3.23e4·9-s − 3.04e4·11-s + 3.23e4·13-s + 1.51e5·15-s + 5.90e5·17-s + 3.46e4·19-s + 1.44e6·21-s − 1.04e6·23-s − 1.50e6·25-s + 2.87e6·27-s − 4.40e6·29-s + 7.40e6·31-s − 6.93e6·33-s + 4.21e6·35-s − 1.02e7·37-s + 7.37e6·39-s + 1.83e7·41-s − 2.52e5·43-s + 2.15e7·45-s + 4.95e7·47-s − 3.10e5·49-s + 1.34e8·51-s + 6.63e7·53-s − 2.02e7·55-s + ⋯
L(s)  = 1  + 1.62·3-s + 0.476·5-s + 0.996·7-s + 1.64·9-s − 0.626·11-s + 0.314·13-s + 0.774·15-s + 1.71·17-s + 0.0610·19-s + 1.61·21-s − 0.781·23-s − 0.772·25-s + 1.04·27-s − 1.15·29-s + 1.43·31-s − 1.01·33-s + 0.474·35-s − 0.897·37-s + 0.510·39-s + 1.01·41-s − 0.0112·43-s + 0.782·45-s + 1.48·47-s − 0.00768·49-s + 2.78·51-s + 1.15·53-s − 0.298·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $1$
Analytic conductor: \(32.9622\)
Root analytic conductor: \(5.74127\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(4.415041922\)
\(L(\frac12)\) \(\approx\) \(4.415041922\)
\(L(\frac{11}{2})\) 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 \( 1 - 76 p T + p^{9} T^{2} \)
5 \( 1 - 666 T + p^{9} T^{2} \)
7 \( 1 - 904 p T + p^{9} T^{2} \)
11 \( 1 + 30420 T + p^{9} T^{2} \)
13 \( 1 - 32338 T + p^{9} T^{2} \)
17 \( 1 - 590994 T + p^{9} T^{2} \)
19 \( 1 - 34676 T + p^{9} T^{2} \)
23 \( 1 + 1048536 T + p^{9} T^{2} \)
29 \( 1 + 4409406 T + p^{9} T^{2} \)
31 \( 1 - 7401184 T + p^{9} T^{2} \)
37 \( 1 + 10234502 T + p^{9} T^{2} \)
41 \( 1 - 18352746 T + p^{9} T^{2} \)
43 \( 1 + 252340 T + p^{9} T^{2} \)
47 \( 1 - 49517136 T + p^{9} T^{2} \)
53 \( 1 - 66396906 T + p^{9} T^{2} \)
59 \( 1 + 61523748 T + p^{9} T^{2} \)
61 \( 1 + 35638622 T + p^{9} T^{2} \)
67 \( 1 - 181742372 T + p^{9} T^{2} \)
71 \( 1 + 90904968 T + p^{9} T^{2} \)
73 \( 1 + 262978678 T + p^{9} T^{2} \)
79 \( 1 - 116502832 T + p^{9} T^{2} \)
83 \( 1 + 9563724 T + p^{9} T^{2} \)
89 \( 1 - 611826714 T + p^{9} T^{2} \)
97 \( 1 + 259312798 T + p^{9} T^{2} \)
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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

−13.45136085031289605027852018344, −12.06397220419177012937520927128, −10.42618468156885971341901507463, −9.435208661257959040006495280388, −8.200346882290701904744620338509, −7.61598482177642003930562340505, −5.55954368154233827535089359857, −3.91179085969037355409123954087, −2.55923696267562455996019129722, −1.44046673806277119764042240767, 1.44046673806277119764042240767, 2.55923696267562455996019129722, 3.91179085969037355409123954087, 5.55954368154233827535089359857, 7.61598482177642003930562340505, 8.200346882290701904744620338509, 9.435208661257959040006495280388, 10.42618468156885971341901507463, 12.06397220419177012937520927128, 13.45136085031289605027852018344

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