Properties

Label 2-2e6-1.1-c11-0-14
Degree $2$
Conductor $64$
Sign $1$
Analytic cond. $49.1739$
Root an. cond. $7.01241$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 516·3-s + 1.05e4·5-s + 4.93e4·7-s + 8.91e4·9-s + 3.09e5·11-s + 1.72e6·13-s + 5.43e6·15-s − 2.27e6·17-s − 4.55e6·19-s + 2.54e7·21-s − 7.28e6·23-s + 6.20e7·25-s − 4.54e7·27-s + 6.90e7·29-s − 1.41e8·31-s + 1.59e8·33-s + 5.19e8·35-s − 7.11e8·37-s + 8.89e8·39-s − 1.22e9·41-s + 3.36e7·43-s + 9.38e8·45-s + 1.23e8·47-s + 4.53e8·49-s − 1.17e9·51-s − 1.10e9·53-s + 3.25e9·55-s + ⋯
L(s)  = 1  + 1.22·3-s + 1.50·5-s + 1.10·7-s + 0.503·9-s + 0.579·11-s + 1.28·13-s + 1.84·15-s − 0.389·17-s − 0.421·19-s + 1.35·21-s − 0.235·23-s + 1.27·25-s − 0.609·27-s + 0.625·29-s − 0.889·31-s + 0.710·33-s + 1.67·35-s − 1.68·37-s + 1.57·39-s − 1.65·41-s + 0.0348·43-s + 0.758·45-s + 0.0783·47-s + 0.229·49-s − 0.477·51-s − 0.363·53-s + 0.872·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $1$
Analytic conductor: \(49.1739\)
Root analytic conductor: \(7.01241\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(5.095357602\)
\(L(\frac12)\) \(\approx\) \(5.095357602\)
\(L(\frac{13}{2})\) 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 \( 1 - 172 p T + p^{11} T^{2} \)
5 \( 1 - 2106 p T + p^{11} T^{2} \)
7 \( 1 - 49304 T + p^{11} T^{2} \)
11 \( 1 - 309420 T + p^{11} T^{2} \)
13 \( 1 - 1723594 T + p^{11} T^{2} \)
17 \( 1 + 2279502 T + p^{11} T^{2} \)
19 \( 1 + 4550444 T + p^{11} T^{2} \)
23 \( 1 + 7282872 T + p^{11} T^{2} \)
29 \( 1 - 69040026 T + p^{11} T^{2} \)
31 \( 1 + 141740704 T + p^{11} T^{2} \)
37 \( 1 + 711366974 T + p^{11} T^{2} \)
41 \( 1 + 1225262214 T + p^{11} T^{2} \)
43 \( 1 - 781540 p T + p^{11} T^{2} \)
47 \( 1 - 123214608 T + p^{11} T^{2} \)
53 \( 1 + 1106121582 T + p^{11} T^{2} \)
59 \( 1 - 9062779932 T + p^{11} T^{2} \)
61 \( 1 - 3854150458 T + p^{11} T^{2} \)
67 \( 1 - 15313764676 T + p^{11} T^{2} \)
71 \( 1 - 20619626328 T + p^{11} T^{2} \)
73 \( 1 + 2063718694 T + p^{11} T^{2} \)
79 \( 1 - 13689871472 T + p^{11} T^{2} \)
83 \( 1 + 65570428908 T + p^{11} T^{2} \)
89 \( 1 + 29715508854 T + p^{11} T^{2} \)
97 \( 1 + 23439626206 T + p^{11} T^{2} \)
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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

−13.04590531259448320816164025089, −11.33834022258209031358172459610, −10.12567158811118820610059134058, −8.906422307387044243899876667934, −8.366936548912076230098707416450, −6.64639225927700146410896491653, −5.32249591245726456227256774612, −3.68363562800224965500083306121, −2.15021801683291206221543622996, −1.47969466187915957762904013773, 1.47969466187915957762904013773, 2.15021801683291206221543622996, 3.68363562800224965500083306121, 5.32249591245726456227256774612, 6.64639225927700146410896491653, 8.366936548912076230098707416450, 8.906422307387044243899876667934, 10.12567158811118820610059134058, 11.33834022258209031358172459610, 13.04590531259448320816164025089

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