Properties

Label 2-2e6-1.1-c21-0-13
Degree $2$
Conductor $64$
Sign $1$
Analytic cond. $178.865$
Root an. cond. $13.3740$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.67e5·3-s − 3.35e6·5-s − 7.07e8·7-s + 1.76e10·9-s + 8.75e10·11-s + 7.41e11·13-s + 5.62e11·15-s − 6.82e12·17-s + 5.17e13·19-s + 1.18e14·21-s + 3.13e14·23-s − 4.65e14·25-s − 1.20e15·27-s − 1.46e15·29-s + 6.42e15·31-s − 1.46e16·33-s + 2.37e15·35-s + 6.93e15·37-s − 1.24e17·39-s + 3.25e16·41-s + 4.37e16·43-s − 5.92e16·45-s + 5.34e16·47-s − 5.75e16·49-s + 1.14e18·51-s + 1.19e18·53-s − 2.93e17·55-s + ⋯
L(s)  = 1  − 1.63·3-s − 0.153·5-s − 0.947·7-s + 1.68·9-s + 1.01·11-s + 1.49·13-s + 0.251·15-s − 0.821·17-s + 1.93·19-s + 1.55·21-s + 1.57·23-s − 0.976·25-s − 1.12·27-s − 0.646·29-s + 1.40·31-s − 1.66·33-s + 0.145·35-s + 0.237·37-s − 2.44·39-s + 0.378·41-s + 0.308·43-s − 0.259·45-s + 0.148·47-s − 0.103·49-s + 1.34·51-s + 0.936·53-s − 0.156·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $1$
Analytic conductor: \(178.865\)
Root analytic conductor: \(13.3740\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(1.321927922\)
\(L(\frac12)\) \(\approx\) \(1.321927922\)
\(L(\frac{23}{2})\) 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 \( 1 + 1.67e5T + 1.04e10T^{2} \)
5 \( 1 + 3.35e6T + 4.76e14T^{2} \)
7 \( 1 + 7.07e8T + 5.58e17T^{2} \)
11 \( 1 - 8.75e10T + 7.40e21T^{2} \)
13 \( 1 - 7.41e11T + 2.47e23T^{2} \)
17 \( 1 + 6.82e12T + 6.90e25T^{2} \)
19 \( 1 - 5.17e13T + 7.14e26T^{2} \)
23 \( 1 - 3.13e14T + 3.94e28T^{2} \)
29 \( 1 + 1.46e15T + 5.13e30T^{2} \)
31 \( 1 - 6.42e15T + 2.08e31T^{2} \)
37 \( 1 - 6.93e15T + 8.55e32T^{2} \)
41 \( 1 - 3.25e16T + 7.38e33T^{2} \)
43 \( 1 - 4.37e16T + 2.00e34T^{2} \)
47 \( 1 - 5.34e16T + 1.30e35T^{2} \)
53 \( 1 - 1.19e18T + 1.62e36T^{2} \)
59 \( 1 + 4.48e17T + 1.54e37T^{2} \)
61 \( 1 - 9.05e17T + 3.10e37T^{2} \)
67 \( 1 + 6.06e17T + 2.22e38T^{2} \)
71 \( 1 - 2.18e19T + 7.52e38T^{2} \)
73 \( 1 + 6.65e19T + 1.34e39T^{2} \)
79 \( 1 - 1.81e18T + 7.08e39T^{2} \)
83 \( 1 + 2.58e20T + 1.99e40T^{2} \)
89 \( 1 + 1.80e20T + 8.65e40T^{2} \)
97 \( 1 - 4.09e20T + 5.27e41T^{2} \)
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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

−11.27937878089901599175919779127, −9.998434573626815520573343360860, −8.985130828071700695340072258055, −7.17515605261103101122261309253, −6.34673323524052930405137245176, −5.62514885301899606820011858812, −4.32896460018308725958145508280, −3.25845603216450371752625366986, −1.26462427732856764854179250856, −0.63742141272551716310797519248, 0.63742141272551716310797519248, 1.26462427732856764854179250856, 3.25845603216450371752625366986, 4.32896460018308725958145508280, 5.62514885301899606820011858812, 6.34673323524052930405137245176, 7.17515605261103101122261309253, 8.985130828071700695340072258055, 9.998434573626815520573343360860, 11.27937878089901599175919779127

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