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Properties

Label 2-2e6-16.5-c11-0-13
Degree $2$
Conductor $64$
Sign $0.443 - 0.896i$
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  + (−442. + 442. i)3-s + (2.49e3 + 2.49e3i)5-s + 3.93e4i·7-s − 2.13e5i·9-s + (3.11e5 + 3.11e5i)11-s + (1.47e6 − 1.47e6i)13-s − 2.20e6·15-s + 6.43e6·17-s + (3.94e6 − 3.94e6i)19-s + (−1.74e7 − 1.74e7i)21-s − 2.40e7i·23-s − 3.63e7i·25-s + (1.61e7 + 1.61e7i)27-s + (1.10e8 − 1.10e8i)29-s + 2.26e8·31-s + ⋯
L(s)  = 1  + (−1.05 + 1.05i)3-s + (0.357 + 0.357i)5-s + 0.885i·7-s − 1.20i·9-s + (0.582 + 0.582i)11-s + (1.09 − 1.09i)13-s − 0.750·15-s + 1.10·17-s + (0.365 − 0.365i)19-s + (−0.930 − 0.930i)21-s − 0.779i·23-s − 0.744i·25-s + (0.216 + 0.216i)27-s + (0.996 − 0.996i)29-s + 1.42·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.443 - 0.896i$
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: \((2,\ 64,\ (\ :11/2),\ 0.443 - 0.896i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.53758 + 0.954532i\)
\(L(\frac12)\) \(\approx\) \(1.53758 + 0.954532i\)
\(L(\frac{13}{2})\) 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 \( 1 + (442. - 442. i)T - 1.77e5iT^{2} \)
5 \( 1 + (-2.49e3 - 2.49e3i)T + 4.88e7iT^{2} \)
7 \( 1 - 3.93e4iT - 1.97e9T^{2} \)
11 \( 1 + (-3.11e5 - 3.11e5i)T + 2.85e11iT^{2} \)
13 \( 1 + (-1.47e6 + 1.47e6i)T - 1.79e12iT^{2} \)
17 \( 1 - 6.43e6T + 3.42e13T^{2} \)
19 \( 1 + (-3.94e6 + 3.94e6i)T - 1.16e14iT^{2} \)
23 \( 1 + 2.40e7iT - 9.52e14T^{2} \)
29 \( 1 + (-1.10e8 + 1.10e8i)T - 1.22e16iT^{2} \)
31 \( 1 - 2.26e8T + 2.54e16T^{2} \)
37 \( 1 + (2.38e8 + 2.38e8i)T + 1.77e17iT^{2} \)
41 \( 1 - 1.45e9iT - 5.50e17T^{2} \)
43 \( 1 + (1.31e8 + 1.31e8i)T + 9.29e17iT^{2} \)
47 \( 1 - 2.03e9T + 2.47e18T^{2} \)
53 \( 1 + (-1.22e9 - 1.22e9i)T + 9.26e18iT^{2} \)
59 \( 1 + (2.81e9 + 2.81e9i)T + 3.01e19iT^{2} \)
61 \( 1 + (5.47e8 - 5.47e8i)T - 4.35e19iT^{2} \)
67 \( 1 + (-1.17e10 + 1.17e10i)T - 1.22e20iT^{2} \)
71 \( 1 - 1.29e9iT - 2.31e20T^{2} \)
73 \( 1 - 2.25e10iT - 3.13e20T^{2} \)
79 \( 1 + 3.10e10T + 7.47e20T^{2} \)
83 \( 1 + (3.09e10 - 3.09e10i)T - 1.28e21iT^{2} \)
89 \( 1 + 7.63e10iT - 2.77e21T^{2} \)
97 \( 1 - 5.73e10T + 7.15e21T^{2} \)
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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

−12.43927798010025179851203711931, −11.59272311633186320655165139139, −10.42917491138221215205646903104, −9.768531986255976561105184207528, −8.328438762867546658844645443520, −6.34843835910258938339388116616, −5.60097862153091303443025345495, −4.37261878608620056070272710274, −2.83128709609929368851034396012, −0.827211808814462906395295618004, 0.971951151622529216139932120437, 1.37376782793385287848783847090, 3.72572526532697259049435886410, 5.41713953003879559730262016041, 6.42214638719194227754634523907, 7.36149251117303458005724279520, 8.846343889472693384215324613732, 10.38449334984693440017454130468, 11.52057347992092375800358337954, 12.24434206844909744437614374379

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