Properties

Degree 2
Conductor $ 2^{4} $
Sign $0.999 + 0.0249i$
Motivic weight 11
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−45.0 − 4.71i)2-s + (396. + 396. i)3-s + (2.00e3 + 424. i)4-s + (7.61e3 − 7.61e3i)5-s + (−1.59e4 − 1.97e4i)6-s + 1.64e4i·7-s + (−8.81e4 − 2.85e4i)8-s + 1.37e5i·9-s + (−3.78e5 + 3.06e5i)10-s + (5.68e5 − 5.68e5i)11-s + (6.26e5 + 9.63e5i)12-s + (−1.11e6 − 1.11e6i)13-s + (7.77e4 − 7.42e5i)14-s + 6.04e6·15-s + (3.83e6 + 1.70e6i)16-s + 6.34e6·17-s + ⋯
L(s)  = 1  + (−0.994 − 0.104i)2-s + (0.942 + 0.942i)3-s + (0.978 + 0.207i)4-s + (1.08 − 1.08i)5-s + (−0.839 − 1.03i)6-s + 0.370i·7-s + (−0.951 − 0.308i)8-s + 0.776i·9-s + (−1.19 + 0.970i)10-s + (1.06 − 1.06i)11-s + (0.726 + 1.11i)12-s + (−0.830 − 0.830i)13-s + (0.0386 − 0.368i)14-s + 2.05·15-s + (0.914 + 0.405i)16-s + 1.08·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0249i)\, \overline{\Lambda}(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0249i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $0.999 + 0.0249i$
motivic weight  =  \(11\)
character  :  $\chi_{16} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ 0.999 + 0.0249i)$
$L(6)$  $\approx$  $1.96004 - 0.0244316i$
$L(\frac12)$  $\approx$  $1.96004 - 0.0244316i$
$L(\frac{13}{2})$   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 \( 1 + (45.0 + 4.71i)T \)
good3 \( 1 + (-396. - 396. i)T + 1.77e5iT^{2} \)
5 \( 1 + (-7.61e3 + 7.61e3i)T - 4.88e7iT^{2} \)
7 \( 1 - 1.64e4iT - 1.97e9T^{2} \)
11 \( 1 + (-5.68e5 + 5.68e5i)T - 2.85e11iT^{2} \)
13 \( 1 + (1.11e6 + 1.11e6i)T + 1.79e12iT^{2} \)
17 \( 1 - 6.34e6T + 3.42e13T^{2} \)
19 \( 1 + (-3.59e6 - 3.59e6i)T + 1.16e14iT^{2} \)
23 \( 1 - 4.79e7iT - 9.52e14T^{2} \)
29 \( 1 + (2.69e7 + 2.69e7i)T + 1.22e16iT^{2} \)
31 \( 1 - 1.46e8T + 2.54e16T^{2} \)
37 \( 1 + (4.69e8 - 4.69e8i)T - 1.77e17iT^{2} \)
41 \( 1 - 3.61e8iT - 5.50e17T^{2} \)
43 \( 1 + (-5.36e8 + 5.36e8i)T - 9.29e17iT^{2} \)
47 \( 1 + 1.46e9T + 2.47e18T^{2} \)
53 \( 1 + (2.55e8 - 2.55e8i)T - 9.26e18iT^{2} \)
59 \( 1 + (2.13e9 - 2.13e9i)T - 3.01e19iT^{2} \)
61 \( 1 + (4.43e9 + 4.43e9i)T + 4.35e19iT^{2} \)
67 \( 1 + (7.43e9 + 7.43e9i)T + 1.22e20iT^{2} \)
71 \( 1 - 8.28e9iT - 2.31e20T^{2} \)
73 \( 1 - 1.36e10iT - 3.13e20T^{2} \)
79 \( 1 + 6.96e8T + 7.47e20T^{2} \)
83 \( 1 + (3.69e10 + 3.69e10i)T + 1.28e21iT^{2} \)
89 \( 1 - 9.40e10iT - 2.77e21T^{2} \)
97 \( 1 + 1.46e11T + 7.15e21T^{2} \)
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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

−16.60147707571934069686433899428, −15.32414354856966706399502576210, −13.90980554398459824948298729069, −12.07035517021371870281243720380, −9.970365556458736312537173126254, −9.314313245801512164226447278519, −8.262897003975823348617701716406, −5.64364091618295353377934750596, −3.16842454192963786071795055491, −1.27223599820369509130419167065, 1.58206479821473094698225881433, 2.60583960541079599202344911317, 6.63303869149266960876922649260, 7.35714739193971589650686524733, 9.203534545511390384829197685861, 10.29621869476971552550179763820, 12.23387120490483239571145757761, 14.19407548007267126162825692815, 14.60988894895298345233169458181, 16.91535954080981264210966020826

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