Properties

Degree 2
Conductor $ 2^{4} $
Sign $0.499 + 0.866i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.460 − 2.79i)2-s + (0.756 − 0.756i)3-s + (−7.57 − 2.57i)4-s + (8.22 + 8.22i)5-s + (−1.76 − 2.46i)6-s + 2.67i·7-s + (−10.6 + 19.9i)8-s + 25.8i·9-s + (26.7 − 19.1i)10-s + (−45.2 − 45.2i)11-s + (−7.67 + 3.78i)12-s + (35.3 − 35.3i)13-s + (7.45 + 1.23i)14-s + 12.4·15-s + (50.7 + 38.9i)16-s − 72.4·17-s + ⋯
L(s)  = 1  + (0.162 − 0.986i)2-s + (0.145 − 0.145i)3-s + (−0.946 − 0.321i)4-s + (0.735 + 0.735i)5-s + (−0.119 − 0.167i)6-s + 0.144i·7-s + (−0.471 + 0.881i)8-s + 0.957i·9-s + (0.845 − 0.605i)10-s + (−1.23 − 1.23i)11-s + (−0.184 + 0.0910i)12-s + (0.755 − 0.755i)13-s + (0.142 + 0.0235i)14-s + 0.214·15-s + (0.793 + 0.609i)16-s − 1.03·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16\)    =    \(2^{4}\)
\( \varepsilon \)  =  $0.499 + 0.866i$
motivic weight  =  \(3\)
character  :  $\chi_{16} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 16,\ (\ :3/2),\ 0.499 + 0.866i)$
$L(2)$  $\approx$  $0.943664 - 0.545046i$
$L(\frac12)$  $\approx$  $0.943664 - 0.545046i$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 2. If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + (-0.460 + 2.79i)T \)
good3 \( 1 + (-0.756 + 0.756i)T - 27iT^{2} \)
5 \( 1 + (-8.22 - 8.22i)T + 125iT^{2} \)
7 \( 1 - 2.67iT - 343T^{2} \)
11 \( 1 + (45.2 + 45.2i)T + 1.33e3iT^{2} \)
13 \( 1 + (-35.3 + 35.3i)T - 2.19e3iT^{2} \)
17 \( 1 + 72.4T + 4.91e3T^{2} \)
19 \( 1 + (-19.4 + 19.4i)T - 6.85e3iT^{2} \)
23 \( 1 - 139. iT - 1.21e4T^{2} \)
29 \( 1 + (-66.0 + 66.0i)T - 2.43e4iT^{2} \)
31 \( 1 - 188.T + 2.97e4T^{2} \)
37 \( 1 + (84.0 + 84.0i)T + 5.06e4iT^{2} \)
41 \( 1 + 104. iT - 6.89e4T^{2} \)
43 \( 1 + (31.4 + 31.4i)T + 7.95e4iT^{2} \)
47 \( 1 + 488.T + 1.03e5T^{2} \)
53 \( 1 + (-149. - 149. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-284. - 284. i)T + 2.05e5iT^{2} \)
61 \( 1 + (228. - 228. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-139. + 139. i)T - 3.00e5iT^{2} \)
71 \( 1 - 453. iT - 3.57e5T^{2} \)
73 \( 1 - 259. iT - 3.89e5T^{2} \)
79 \( 1 - 323.T + 4.93e5T^{2} \)
83 \( 1 + (563. - 563. i)T - 5.71e5iT^{2} \)
89 \( 1 + 866. iT - 7.04e5T^{2} \)
97 \( 1 + 936.T + 9.12e5T^{2} \)
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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

−18.58524497962152426945672299033, −17.72158127192754193792343281251, −15.65799274584242342695014859645, −13.72613266147089125675496281328, −13.35846570225670564111675053452, −11.15711219352142456013806911767, −10.30085995814503385035996984795, −8.357650192784133268869138289171, −5.60259368821300895672858813444, −2.72333088463365132349406283853, 4.70959041005156714992572015765, 6.58253413057200100532441752179, 8.599453103531040995629116943758, 9.848809066563963705704552006510, 12.54250575719246046910133861827, 13.59133842211992691008963412539, 15.06248374774745871847382616081, 16.17017258709217705895260223014, 17.52471607872774941646223991895, 18.30966085760113228255287537381

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