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Properties

Label 2-2e6-8.5-c21-0-31
Degree $2$
Conductor $64$
Sign $0.258 + 0.965i$
Analytic cond. $178.865$
Root an. cond. $13.3740$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.96e3i·3-s − 1.30e7i·5-s + 3.86e8·7-s + 1.04e10·9-s − 9.89e10i·11-s + 7.91e11i·13-s + 5.17e10·15-s − 7.19e12·17-s − 4.44e13i·19-s + 1.52e12i·21-s + 3.12e14·23-s + 3.06e14·25-s + 8.28e13i·27-s − 3.52e15i·29-s − 3.23e15·31-s + ⋯
L(s)  = 1  + 0.0387i·3-s − 0.598i·5-s + 0.516·7-s + 0.998·9-s − 1.14i·11-s + 1.59i·13-s + 0.0231·15-s − 0.865·17-s − 1.66i·19-s + 0.0200i·21-s + 1.57·23-s + 0.642·25-s + 0.0774i·27-s − 1.55i·29-s − 0.709·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.258 + 0.965i$
Analytic conductor: \(178.865\)
Root analytic conductor: \(13.3740\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :21/2),\ 0.258 + 0.965i)\)

Particular Values

\(L(11)\) \(\approx\) \(2.680545019\)
\(L(\frac12)\) \(\approx\) \(2.680545019\)
\(L(\frac{23}{2})\) 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 \( 1 - 3.96e3iT - 1.04e10T^{2} \)
5 \( 1 + 1.30e7iT - 4.76e14T^{2} \)
7 \( 1 - 3.86e8T + 5.58e17T^{2} \)
11 \( 1 + 9.89e10iT - 7.40e21T^{2} \)
13 \( 1 - 7.91e11iT - 2.47e23T^{2} \)
17 \( 1 + 7.19e12T + 6.90e25T^{2} \)
19 \( 1 + 4.44e13iT - 7.14e26T^{2} \)
23 \( 1 - 3.12e14T + 3.94e28T^{2} \)
29 \( 1 + 3.52e15iT - 5.13e30T^{2} \)
31 \( 1 + 3.23e15T + 2.08e31T^{2} \)
37 \( 1 - 2.22e16iT - 8.55e32T^{2} \)
41 \( 1 - 1.29e17T + 7.38e33T^{2} \)
43 \( 1 - 1.74e17iT - 2.00e34T^{2} \)
47 \( 1 - 4.14e17T + 1.30e35T^{2} \)
53 \( 1 + 5.90e17iT - 1.62e36T^{2} \)
59 \( 1 - 1.12e18iT - 1.54e37T^{2} \)
61 \( 1 - 3.51e18iT - 3.10e37T^{2} \)
67 \( 1 + 1.30e19iT - 2.22e38T^{2} \)
71 \( 1 + 3.88e19T + 7.52e38T^{2} \)
73 \( 1 + 7.31e19T + 1.34e39T^{2} \)
79 \( 1 - 1.02e20T + 7.08e39T^{2} \)
83 \( 1 - 4.06e19iT - 1.99e40T^{2} \)
89 \( 1 - 7.66e19T + 8.65e40T^{2} \)
97 \( 1 + 1.36e19T + 5.27e41T^{2} \)
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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

−10.95804282945716153920087325617, −9.285444535631123134552668692621, −8.795682043681809405413992318077, −7.33954660634505477068565275031, −6.37475031357149903410612030971, −4.81409488747327032380995346070, −4.28821148174455807173597944206, −2.67534928699499892316173122231, −1.42964377361309922006557185331, −0.56984017437869253187892038705, 1.01351537818109205768266939357, 2.00593211634536533939658697093, 3.25072693258592454436858127422, 4.46157433555141325314866232922, 5.53312202269985618870235922546, 7.02346398891475406187967947759, 7.59533308741453722730482135013, 9.011090112629054176423591134915, 10.35433524771862134105343146511, 10.79950866033290679321697525513

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