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Properties

Label 2-2e6-8.5-c21-0-40
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  − 1.13e5i·3-s − 2.14e7i·5-s − 5.47e8·7-s − 2.31e9·9-s − 1.01e11i·11-s − 4.95e11i·13-s − 2.42e12·15-s − 1.25e13·17-s − 2.73e12i·19-s + 6.19e13i·21-s − 1.32e14·23-s + 1.81e13·25-s − 9.20e14i·27-s − 2.72e15i·29-s + 4.42e15·31-s + ⋯
L(s)  = 1  − 1.10i·3-s − 0.980i·5-s − 0.733·7-s − 0.221·9-s − 1.17i·11-s − 0.997i·13-s − 1.08·15-s − 1.50·17-s − 0.102i·19-s + 0.810i·21-s − 0.666·23-s + 0.0379·25-s − 0.860i·27-s − 1.20i·29-s + 0.969·31-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

\(L(11)\) \(\approx\) \(1.058873551\)
\(L(\frac12)\) \(\approx\) \(1.058873551\)
\(L(\frac{23}{2})\) 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 \( 1 + 1.13e5iT - 1.04e10T^{2} \)
5 \( 1 + 2.14e7iT - 4.76e14T^{2} \)
7 \( 1 + 5.47e8T + 5.58e17T^{2} \)
11 \( 1 + 1.01e11iT - 7.40e21T^{2} \)
13 \( 1 + 4.95e11iT - 2.47e23T^{2} \)
17 \( 1 + 1.25e13T + 6.90e25T^{2} \)
19 \( 1 + 2.73e12iT - 7.14e26T^{2} \)
23 \( 1 + 1.32e14T + 3.94e28T^{2} \)
29 \( 1 + 2.72e15iT - 5.13e30T^{2} \)
31 \( 1 - 4.42e15T + 2.08e31T^{2} \)
37 \( 1 + 8.33e15iT - 8.55e32T^{2} \)
41 \( 1 + 7.49e16T + 7.38e33T^{2} \)
43 \( 1 - 2.77e16iT - 2.00e34T^{2} \)
47 \( 1 + 1.56e17T + 1.30e35T^{2} \)
53 \( 1 + 6.67e17iT - 1.62e36T^{2} \)
59 \( 1 + 5.80e18iT - 1.54e37T^{2} \)
61 \( 1 - 9.87e18iT - 3.10e37T^{2} \)
67 \( 1 + 2.22e19iT - 2.22e38T^{2} \)
71 \( 1 + 2.93e19T + 7.52e38T^{2} \)
73 \( 1 - 3.26e19T + 1.34e39T^{2} \)
79 \( 1 + 5.86e19T + 7.08e39T^{2} \)
83 \( 1 + 9.06e19iT - 1.99e40T^{2} \)
89 \( 1 + 1.01e20T + 8.65e40T^{2} \)
97 \( 1 + 6.59e20T + 5.27e41T^{2} \)
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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

−9.983681646327220408952311358719, −8.689893536228831592170696745412, −7.996216561641494973811665792771, −6.64538229961058561360934897162, −5.87450155431005419114072329103, −4.50129529033737754008409523684, −3.09111655909512450398806708650, −1.91085781199514597302851625766, −0.68368859301202327769900612562, −0.27202146851860711701964055059, 1.81108071874516937069903400723, 2.93512364844751241830483378186, 4.03368313024550826175169058692, 4.79792561676839709393635962758, 6.49825383211863945263009855398, 7.07536606305293914239994239815, 8.874405551974280393992849241652, 9.816313479540433286386005506328, 10.45584393960414750786150301724, 11.50454936918795114002799724207

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