Properties

Degree 2
Conductor $ 2^{4} $
Sign $0.908 - 0.418i$
Motivic weight 11
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.445 − 45.2i)2-s + (−296. − 296. i)3-s + (−2.04e3 + 40.3i)4-s + (−2.86e3 + 2.86e3i)5-s + (−1.32e4 + 1.35e4i)6-s + 8.38e3i·7-s + (2.73e3 + 9.26e4i)8-s − 1.24e3i·9-s + (1.31e5 + 1.28e5i)10-s + (8.45e4 − 8.45e4i)11-s + (6.19e5 + 5.95e5i)12-s + (−3.85e5 − 3.85e5i)13-s + (3.79e5 − 3.73e3i)14-s + 1.70e6·15-s + (4.19e6 − 1.65e5i)16-s + 1.56e6·17-s + ⋯
L(s)  = 1  + (−0.00985 − 0.999i)2-s + (−0.704 − 0.704i)3-s + (−0.999 + 0.0197i)4-s + (−0.410 + 0.410i)5-s + (−0.697 + 0.711i)6-s + 0.188i·7-s + (0.0295 + 0.999i)8-s − 0.00701i·9-s + (0.414 + 0.406i)10-s + (0.158 − 0.158i)11-s + (0.718 + 0.690i)12-s + (−0.287 − 0.287i)13-s + (0.188 − 0.00185i)14-s + 0.578·15-s + (0.999 − 0.0393i)16-s + 0.267·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $0.908 - 0.418i$
motivic weight  =  \(11\)
character  :  $\chi_{16} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :11/2),\ 0.908 - 0.418i)$
$L(6)$  $\approx$  $0.516035 + 0.113259i$
$L(\frac12)$  $\approx$  $0.516035 + 0.113259i$
$L(\frac{13}{2})$   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 \( 1 + (0.445 + 45.2i)T \)
good3 \( 1 + (296. + 296. i)T + 1.77e5iT^{2} \)
5 \( 1 + (2.86e3 - 2.86e3i)T - 4.88e7iT^{2} \)
7 \( 1 - 8.38e3iT - 1.97e9T^{2} \)
11 \( 1 + (-8.45e4 + 8.45e4i)T - 2.85e11iT^{2} \)
13 \( 1 + (3.85e5 + 3.85e5i)T + 1.79e12iT^{2} \)
17 \( 1 - 1.56e6T + 3.42e13T^{2} \)
19 \( 1 + (-9.59e6 - 9.59e6i)T + 1.16e14iT^{2} \)
23 \( 1 - 3.88e7iT - 9.52e14T^{2} \)
29 \( 1 + (8.81e7 + 8.81e7i)T + 1.22e16iT^{2} \)
31 \( 1 + 7.23e7T + 2.54e16T^{2} \)
37 \( 1 + (9.98e7 - 9.98e7i)T - 1.77e17iT^{2} \)
41 \( 1 - 1.13e9iT - 5.50e17T^{2} \)
43 \( 1 + (1.05e9 - 1.05e9i)T - 9.29e17iT^{2} \)
47 \( 1 - 1.33e9T + 2.47e18T^{2} \)
53 \( 1 + (-2.50e9 + 2.50e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (3.81e9 - 3.81e9i)T - 3.01e19iT^{2} \)
61 \( 1 + (7.73e9 + 7.73e9i)T + 4.35e19iT^{2} \)
67 \( 1 + (-2.91e9 - 2.91e9i)T + 1.22e20iT^{2} \)
71 \( 1 - 1.77e10iT - 2.31e20T^{2} \)
73 \( 1 + 2.04e10iT - 3.13e20T^{2} \)
79 \( 1 + 3.60e10T + 7.47e20T^{2} \)
83 \( 1 + (-5.73e9 - 5.73e9i)T + 1.28e21iT^{2} \)
89 \( 1 - 1.07e10iT - 2.77e21T^{2} \)
97 \( 1 + 7.36e9T + 7.15e21T^{2} \)
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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

−16.97156541770474887490289175267, −14.92839582372962901984175659025, −13.34338183821402336810313281905, −12.05800797124869255085591836680, −11.30700166402161341798461890141, −9.634261958061048135789423092094, −7.63538423665062778459008005224, −5.63285779348746706759708832484, −3.42704056327955950933595966651, −1.33525155026203846050241903558, 0.28392011587841274389626981668, 4.24631732695354033421482600245, 5.39540946944022208345122584163, 7.23368119030262545779524439356, 8.939516341533194257381383548911, 10.46712542735454546670587378927, 12.22375079498930424257456072680, 13.89615660079523293474845923694, 15.37385063583400437894278225823, 16.37555960911181973586907155315

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