Properties

Degree 2
Conductor $ 2^{4} $
Sign $-0.983 + 0.181i$
Motivic weight 11
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−33.6 + 30.2i)2-s + (−4.32 − 4.32i)3-s + (214. − 2.03e3i)4-s + (2.23e3 − 2.23e3i)5-s + (276. + 14.5i)6-s + 7.14e4i·7-s + (5.44e4 + 7.50e4i)8-s − 1.77e5i·9-s + (−7.50e3 + 1.42e5i)10-s + (−2.36e5 + 2.36e5i)11-s + (−9.74e3 + 7.88e3i)12-s + (−282. − 282. i)13-s + (−2.16e6 − 2.40e6i)14-s − 1.93e4·15-s + (−4.10e6 − 8.75e5i)16-s − 8.99e6·17-s + ⋯
L(s)  = 1  + (−0.743 + 0.668i)2-s + (−0.0102 − 0.0102i)3-s + (0.104 − 0.994i)4-s + (0.319 − 0.319i)5-s + (0.0145 + 0.000763i)6-s + 1.60i·7-s + (0.587 + 0.809i)8-s − 0.999i·9-s + (−0.0237 + 0.451i)10-s + (−0.442 + 0.442i)11-s + (−0.0113 + 0.00914i)12-s + (−0.000211 − 0.000211i)13-s + (−1.07 − 1.19i)14-s − 0.00657·15-s + (−0.977 − 0.208i)16-s − 1.53·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $-0.983 + 0.181i$
motivic weight  =  \(11\)
character  :  $\chi_{16} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ -0.983 + 0.181i)$
$L(6)$  $\approx$  $0.0302753 - 0.330831i$
$L(\frac12)$  $\approx$  $0.0302753 - 0.330831i$
$L(\frac{13}{2})$   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 \( 1 + (33.6 - 30.2i)T \)
good3 \( 1 + (4.32 + 4.32i)T + 1.77e5iT^{2} \)
5 \( 1 + (-2.23e3 + 2.23e3i)T - 4.88e7iT^{2} \)
7 \( 1 - 7.14e4iT - 1.97e9T^{2} \)
11 \( 1 + (2.36e5 - 2.36e5i)T - 2.85e11iT^{2} \)
13 \( 1 + (282. + 282. i)T + 1.79e12iT^{2} \)
17 \( 1 + 8.99e6T + 3.42e13T^{2} \)
19 \( 1 + (1.07e7 + 1.07e7i)T + 1.16e14iT^{2} \)
23 \( 1 - 3.55e7iT - 9.52e14T^{2} \)
29 \( 1 + (4.42e7 + 4.42e7i)T + 1.22e16iT^{2} \)
31 \( 1 + 1.07e8T + 2.54e16T^{2} \)
37 \( 1 + (4.45e8 - 4.45e8i)T - 1.77e17iT^{2} \)
41 \( 1 + 1.29e8iT - 5.50e17T^{2} \)
43 \( 1 + (5.58e8 - 5.58e8i)T - 9.29e17iT^{2} \)
47 \( 1 - 4.62e8T + 2.47e18T^{2} \)
53 \( 1 + (-1.27e9 + 1.27e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (-5.52e9 + 5.52e9i)T - 3.01e19iT^{2} \)
61 \( 1 + (1.52e9 + 1.52e9i)T + 4.35e19iT^{2} \)
67 \( 1 + (-1.43e10 - 1.43e10i)T + 1.22e20iT^{2} \)
71 \( 1 + 7.48e9iT - 2.31e20T^{2} \)
73 \( 1 + 1.59e10iT - 3.13e20T^{2} \)
79 \( 1 - 6.06e9T + 7.47e20T^{2} \)
83 \( 1 + (-8.65e9 - 8.65e9i)T + 1.28e21iT^{2} \)
89 \( 1 - 5.51e10iT - 2.77e21T^{2} \)
97 \( 1 + 1.47e11T + 7.15e21T^{2} \)
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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

−17.38522808280207098720984089604, −15.49770739867957597726071785516, −15.15074040992881598352742877518, −13.07372548049865690139465572202, −11.42506099400731419327868386657, −9.453236148041570762670446147393, −8.691366758585365058696879245027, −6.64065087756086824996318209627, −5.26396281767787804654736777644, −2.05690934696536456627644567071, 0.17161865364580113431742170449, 2.11926989848912297702861488982, 4.13596677489869127842288329087, 6.98518429303744164955655447377, 8.398362464680209768365910691322, 10.44961803414816010476653155554, 10.80015904528570267748992046512, 12.95413645750264935733773946033, 13.97743538200002323384199758418, 16.26924082554054802892542568345

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