Properties

Degree 2
Conductor $ 2^{4} $
Sign $0.991 + 0.129i$
Motivic weight 11
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.89 + 44.8i)2-s + (137. + 137. i)3-s + (−1.97e3 − 529. i)4-s + (3.89e3 − 3.89e3i)5-s + (−6.95e3 + 5.34e3i)6-s − 3.79e4i·7-s + (3.54e4 − 8.56e4i)8-s − 1.39e5i·9-s + (1.51e5 + 1.97e5i)10-s + (−6.81e4 + 6.81e4i)11-s + (−1.98e5 − 3.43e5i)12-s + (−1.09e5 − 1.09e5i)13-s + (1.70e6 + 2.23e5i)14-s + 1.06e6·15-s + (3.63e6 + 2.09e6i)16-s + 8.85e6·17-s + ⋯
L(s)  = 1  + (−0.130 + 0.991i)2-s + (0.325 + 0.325i)3-s + (−0.966 − 0.258i)4-s + (0.558 − 0.558i)5-s + (−0.365 + 0.280i)6-s − 0.853i·7-s + (0.382 − 0.924i)8-s − 0.787i·9-s + (0.480 + 0.626i)10-s + (−0.127 + 0.127i)11-s + (−0.230 − 0.398i)12-s + (−0.0820 − 0.0820i)13-s + (0.845 + 0.111i)14-s + 0.363·15-s + (0.866 + 0.499i)16-s + 1.51·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $0.991 + 0.129i$
motivic weight  =  \(11\)
character  :  $\chi_{16} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ 0.991 + 0.129i)$
$L(6)$  $\approx$  $1.67338 - 0.109000i$
$L(\frac12)$  $\approx$  $1.67338 - 0.109000i$
$L(\frac{13}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p(T)\) is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (5.89 - 44.8i)T \)
good3 \( 1 + (-137. - 137. i)T + 1.77e5iT^{2} \)
5 \( 1 + (-3.89e3 + 3.89e3i)T - 4.88e7iT^{2} \)
7 \( 1 + 3.79e4iT - 1.97e9T^{2} \)
11 \( 1 + (6.81e4 - 6.81e4i)T - 2.85e11iT^{2} \)
13 \( 1 + (1.09e5 + 1.09e5i)T + 1.79e12iT^{2} \)
17 \( 1 - 8.85e6T + 3.42e13T^{2} \)
19 \( 1 + (7.12e5 + 7.12e5i)T + 1.16e14iT^{2} \)
23 \( 1 + 2.69e7iT - 9.52e14T^{2} \)
29 \( 1 + (1.10e8 + 1.10e8i)T + 1.22e16iT^{2} \)
31 \( 1 + 1.15e8T + 2.54e16T^{2} \)
37 \( 1 + (-3.15e8 + 3.15e8i)T - 1.77e17iT^{2} \)
41 \( 1 + 8.66e8iT - 5.50e17T^{2} \)
43 \( 1 + (3.13e8 - 3.13e8i)T - 9.29e17iT^{2} \)
47 \( 1 - 1.93e9T + 2.47e18T^{2} \)
53 \( 1 + (-8.24e8 + 8.24e8i)T - 9.26e18iT^{2} \)
59 \( 1 + (7.47e9 - 7.47e9i)T - 3.01e19iT^{2} \)
61 \( 1 + (6.44e8 + 6.44e8i)T + 4.35e19iT^{2} \)
67 \( 1 + (-5.39e9 - 5.39e9i)T + 1.22e20iT^{2} \)
71 \( 1 - 2.90e10iT - 2.31e20T^{2} \)
73 \( 1 + 1.12e10iT - 3.13e20T^{2} \)
79 \( 1 + 2.03e7T + 7.47e20T^{2} \)
83 \( 1 + (-2.10e10 - 2.10e10i)T + 1.28e21iT^{2} \)
89 \( 1 + 7.01e10iT - 2.77e21T^{2} \)
97 \( 1 + 3.85e10T + 7.15e21T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.52358591687886153506047682267, −15.03873970101092484049340590028, −13.99567835720548556805382785558, −12.68900889733888213135763953025, −10.13178492563260375841808123919, −9.072679932167324688955588087513, −7.46022749603261001413403650379, −5.70530602659023159213176000139, −3.98114831089578837703192349779, −0.794601337149158505272984983294, 1.74171612001673448932990819012, 3.02173120898469245343548382973, 5.44086690756244471472456933560, 7.88617903190498291095874751514, 9.450688628755526302216500705427, 10.78882686668686610563133349056, 12.27128864725359897788301464271, 13.56480543371982971431335775501, 14.64166945115065800686553994946, 16.73749782631497655371729489423

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