Properties

Label 2-2e6-16.5-c11-0-11
Degree $2$
Conductor $64$
Sign $0.878 + 0.477i$
Analytic cond. $49.1739$
Root an. cond. $7.01241$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (145. − 145. i)3-s + (−4.65e3 − 4.65e3i)5-s − 4.32e4i·7-s + 1.35e5i·9-s + (6.85e5 + 6.85e5i)11-s + (−1.40e6 + 1.40e6i)13-s − 1.34e6·15-s + 3.84e6·17-s + (7.81e6 − 7.81e6i)19-s + (−6.27e6 − 6.27e6i)21-s + 1.99e7i·23-s − 5.53e6i·25-s + (4.52e7 + 4.52e7i)27-s + (1.18e8 − 1.18e8i)29-s − 5.32e7·31-s + ⋯
L(s)  = 1  + (0.344 − 0.344i)3-s + (−0.665 − 0.665i)5-s − 0.972i·7-s + 0.762i·9-s + (1.28 + 1.28i)11-s + (−1.04 + 1.04i)13-s − 0.459·15-s + 0.656·17-s + (0.724 − 0.724i)19-s + (−0.335 − 0.335i)21-s + 0.647i·23-s − 0.113i·25-s + (0.607 + 0.607i)27-s + (1.07 − 1.07i)29-s − 0.333·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.878 + 0.477i$
Analytic conductor: \(49.1739\)
Root analytic conductor: \(7.01241\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :11/2),\ 0.878 + 0.477i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.09010 - 0.530941i\)
\(L(\frac12)\) \(\approx\) \(2.09010 - 0.530941i\)
\(L(\frac{13}{2})\) 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 \( 1 + (-145. + 145. i)T - 1.77e5iT^{2} \)
5 \( 1 + (4.65e3 + 4.65e3i)T + 4.88e7iT^{2} \)
7 \( 1 + 4.32e4iT - 1.97e9T^{2} \)
11 \( 1 + (-6.85e5 - 6.85e5i)T + 2.85e11iT^{2} \)
13 \( 1 + (1.40e6 - 1.40e6i)T - 1.79e12iT^{2} \)
17 \( 1 - 3.84e6T + 3.42e13T^{2} \)
19 \( 1 + (-7.81e6 + 7.81e6i)T - 1.16e14iT^{2} \)
23 \( 1 - 1.99e7iT - 9.52e14T^{2} \)
29 \( 1 + (-1.18e8 + 1.18e8i)T - 1.22e16iT^{2} \)
31 \( 1 + 5.32e7T + 2.54e16T^{2} \)
37 \( 1 + (3.70e8 + 3.70e8i)T + 1.77e17iT^{2} \)
41 \( 1 + 8.83e7iT - 5.50e17T^{2} \)
43 \( 1 + (1.81e8 + 1.81e8i)T + 9.29e17iT^{2} \)
47 \( 1 - 1.92e9T + 2.47e18T^{2} \)
53 \( 1 + (-1.39e9 - 1.39e9i)T + 9.26e18iT^{2} \)
59 \( 1 + (-2.09e9 - 2.09e9i)T + 3.01e19iT^{2} \)
61 \( 1 + (-1.23e9 + 1.23e9i)T - 4.35e19iT^{2} \)
67 \( 1 + (-9.73e9 + 9.73e9i)T - 1.22e20iT^{2} \)
71 \( 1 - 2.30e9iT - 2.31e20T^{2} \)
73 \( 1 - 2.18e10iT - 3.13e20T^{2} \)
79 \( 1 - 1.21e8T + 7.47e20T^{2} \)
83 \( 1 + (-4.69e10 + 4.69e10i)T - 1.28e21iT^{2} \)
89 \( 1 + 6.68e10iT - 2.77e21T^{2} \)
97 \( 1 - 4.52e10T + 7.15e21T^{2} \)
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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

−12.37085004586148286047673713570, −11.69762280508497507840491203323, −10.09549193483571919877642773795, −9.036385928636997041294241458138, −7.56461327592760528438641980384, −7.04831210305321671392773624253, −4.82519853881208005067388820336, −4.01083974811806351300152993961, −2.06742629037624753413450868971, −0.833873000883717581749637332557, 0.838386067823435438709941658496, 2.95587303546610643319515759659, 3.59898133360604840402701370767, 5.47139717377476227172726243854, 6.73087542175811628148328741599, 8.190307512273116104804884583572, 9.164485461984816741000699244958, 10.37332139574095970010017227371, 11.78632977182089518542004424612, 12.29788702972983721199550105176

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