↑

Properties

Label 2-2e4-16.3-c8-0-13
Degree $2$
Conductor $16$
Sign $0.663 + 0.748i$
Analytic cond. $6.51805$
Root an. cond. $2.55304$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (15.9 − 1.32i)2-s + (81.1 − 81.1i)3-s + (252. − 42.3i)4-s + (−62.7 + 62.7i)5-s + (1.18e3 − 1.40e3i)6-s − 2.75e3·7-s + (3.96e3 − 1.01e3i)8-s − 6.59e3i·9-s + (−917. + 1.08e3i)10-s + (1.61e4 + 1.61e4i)11-s + (1.70e4 − 2.39e4i)12-s + (−1.23e4 − 1.23e4i)13-s + (−4.39e4 + 3.66e3i)14-s + 1.01e4i·15-s + (6.19e4 − 2.13e4i)16-s + 5.30e4·17-s + ⋯
L(s)  = 1  + (0.996 − 0.0829i)2-s + (1.00 − 1.00i)3-s + (0.986 − 0.165i)4-s + (−0.100 + 0.100i)5-s + (0.914 − 1.08i)6-s − 1.14·7-s + (0.969 − 0.246i)8-s − 1.00i·9-s + (−0.0917 + 0.108i)10-s + (1.10 + 1.10i)11-s + (0.822 − 1.15i)12-s + (−0.433 − 0.433i)13-s + (−1.14 + 0.0952i)14-s + 0.201i·15-s + (0.945 − 0.326i)16-s + 0.635·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.663 + 0.748i$
Analytic conductor: \(6.51805\)
Root analytic conductor: \(2.55304\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :4),\ 0.663 + 0.748i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.18957 - 1.43561i\)
\(L(\frac12)\) \(\approx\) \(3.18957 - 1.43561i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-15.9 + 1.32i)T \)
good3 \( 1 + (-81.1 + 81.1i)T - 6.56e3iT^{2} \)
5 \( 1 + (62.7 - 62.7i)T - 3.90e5iT^{2} \)
7 \( 1 + 2.75e3T + 5.76e6T^{2} \)
11 \( 1 + (-1.61e4 - 1.61e4i)T + 2.14e8iT^{2} \)
13 \( 1 + (1.23e4 + 1.23e4i)T + 8.15e8iT^{2} \)
17 \( 1 - 5.30e4T + 6.97e9T^{2} \)
19 \( 1 + (1.29e5 - 1.29e5i)T - 1.69e10iT^{2} \)
23 \( 1 + 3.38e5T + 7.83e10T^{2} \)
29 \( 1 + (-2.13e5 - 2.13e5i)T + 5.00e11iT^{2} \)
31 \( 1 + 1.67e6iT - 8.52e11T^{2} \)
37 \( 1 + (1.55e5 - 1.55e5i)T - 3.51e12iT^{2} \)
41 \( 1 + 1.43e6iT - 7.98e12T^{2} \)
43 \( 1 + (2.08e6 + 2.08e6i)T + 1.16e13iT^{2} \)
47 \( 1 - 5.05e6iT - 2.38e13T^{2} \)
53 \( 1 + (-7.69e6 + 7.69e6i)T - 6.22e13iT^{2} \)
59 \( 1 + (1.51e7 + 1.51e7i)T + 1.46e14iT^{2} \)
61 \( 1 + (1.51e6 + 1.51e6i)T + 1.91e14iT^{2} \)
67 \( 1 + (-1.50e7 + 1.50e7i)T - 4.06e14iT^{2} \)
71 \( 1 + 1.73e7T + 6.45e14T^{2} \)
73 \( 1 + 1.86e7iT - 8.06e14T^{2} \)
79 \( 1 - 4.84e7iT - 1.51e15T^{2} \)
83 \( 1 + (-8.26e6 + 8.26e6i)T - 2.25e15iT^{2} \)
89 \( 1 - 7.04e6iT - 3.93e15T^{2} \)
97 \( 1 - 8.35e7T + 7.83e15T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.85879562468851727368547121261, −15.14585022046781265568866292789, −14.21441229568909328127014416325, −12.91399014811183762231132947572, −12.20063308330712667687787174246, −9.832612712490766613836320485760, −7.63863473009965357417859667612, −6.39658309621443590641749708466, −3.64319973051875560583077667603, −1.98135089413321593837325223934, 3.00772709927322091714794536393, 4.18640902223892260321496557386, 6.43678315126659922043806740084, 8.719509475674360943407901793725, 10.22147403580578419042531370532, 12.06740545586899292864468590907, 13.70186074593440401795536595000, 14.59364633624912741398532808325, 15.87625680343116803582820563396, 16.61469748975036301317037299763

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