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Properties

Label 2-2e6-8.5-c21-0-29
Degree $2$
Conductor $64$
Sign $-0.965 + 0.258i$
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  − 1.55e5i·3-s − 8.02e6i·5-s − 7.41e8·7-s − 1.36e10·9-s + 5.97e10i·11-s − 4.24e11i·13-s − 1.24e12·15-s + 1.25e13·17-s − 1.26e13i·19-s + 1.15e14i·21-s + 2.08e14·23-s + 4.12e14·25-s + 4.97e14i·27-s − 5.67e14i·29-s + 5.21e15·31-s + ⋯
L(s)  = 1  − 1.51i·3-s − 0.367i·5-s − 0.992·7-s − 1.30·9-s + 0.694i·11-s − 0.853i·13-s − 0.558·15-s + 1.51·17-s − 0.472i·19-s + 1.50i·21-s + 1.04·23-s + 0.864·25-s + 0.464i·27-s − 0.250i·29-s + 1.14·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-0.965 + 0.258i$
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: \((2,\ 64,\ (\ :21/2),\ -0.965 + 0.258i)\)

Particular Values

\(L(11)\) \(\approx\) \(2.072786179\)
\(L(\frac12)\) \(\approx\) \(2.072786179\)
\(L(\frac{23}{2})\) 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 \( 1 + 1.55e5iT - 1.04e10T^{2} \)
5 \( 1 + 8.02e6iT - 4.76e14T^{2} \)
7 \( 1 + 7.41e8T + 5.58e17T^{2} \)
11 \( 1 - 5.97e10iT - 7.40e21T^{2} \)
13 \( 1 + 4.24e11iT - 2.47e23T^{2} \)
17 \( 1 - 1.25e13T + 6.90e25T^{2} \)
19 \( 1 + 1.26e13iT - 7.14e26T^{2} \)
23 \( 1 - 2.08e14T + 3.94e28T^{2} \)
29 \( 1 + 5.67e14iT - 5.13e30T^{2} \)
31 \( 1 - 5.21e15T + 2.08e31T^{2} \)
37 \( 1 + 4.07e16iT - 8.55e32T^{2} \)
41 \( 1 + 2.32e16T + 7.38e33T^{2} \)
43 \( 1 - 1.35e15iT - 2.00e34T^{2} \)
47 \( 1 - 3.31e17T + 1.30e35T^{2} \)
53 \( 1 + 7.09e17iT - 1.62e36T^{2} \)
59 \( 1 - 3.66e18iT - 1.54e37T^{2} \)
61 \( 1 + 7.38e18iT - 3.10e37T^{2} \)
67 \( 1 + 4.59e18iT - 2.22e38T^{2} \)
71 \( 1 - 4.10e19T + 7.52e38T^{2} \)
73 \( 1 + 4.51e19T + 1.34e39T^{2} \)
79 \( 1 - 1.57e20T + 7.08e39T^{2} \)
83 \( 1 + 1.74e20iT - 1.99e40T^{2} \)
89 \( 1 + 2.32e20T + 8.65e40T^{2} \)
97 \( 1 - 8.12e20T + 5.27e41T^{2} \)
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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

−10.39952734568705383447980230366, −9.224062545775011537037721499583, −7.960102716144685389342412685787, −7.15003947356937709493605435349, −6.23036449940639569391764896540, −5.10156327302366386321872728267, −3.32030924021423305553372078043, −2.37667402346182673440783160137, −1.04770564609002644217879576162, −0.52450555803526151594954005580, 0.981671208773898540598205002572, 2.96577214506439002262893582280, 3.44674452900058253032301899378, 4.62467888397012139853877785247, 5.74138897537967004169155554384, 6.85326248654056921486845999253, 8.486142715646929868031167121014, 9.530427365058920030282179957716, 10.17935927248731785033776325468, 11.11388237496692646170868030363

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