↑

Properties

Label 2-2e4-16.5-c7-0-9
Degree $2$
Conductor $16$
Sign $0.904 + 0.426i$
Analytic cond. $4.99816$
Root an. cond. $2.23565$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (11.3 + 0.135i)2-s + (26.1 − 26.1i)3-s + (127. + 3.05i)4-s + (−31.6 − 31.6i)5-s + (299. − 292. i)6-s − 444. i·7-s + (1.44e3 + 51.8i)8-s + 820. i·9-s + (−353. − 361. i)10-s + (472. + 472. i)11-s + (3.42e3 − 3.26e3i)12-s + (−2.74e3 + 2.74e3i)13-s + (60.0 − 5.02e3i)14-s − 1.65e3·15-s + (1.63e4 + 782. i)16-s − 2.79e4·17-s + ⋯
L(s)  = 1  + (0.999 + 0.0119i)2-s + (0.558 − 0.558i)3-s + (0.999 + 0.0238i)4-s + (−0.113 − 0.113i)5-s + (0.565 − 0.552i)6-s − 0.489i·7-s + (0.999 + 0.0358i)8-s + 0.375i·9-s + (−0.111 − 0.114i)10-s + (0.107 + 0.107i)11-s + (0.572 − 0.545i)12-s + (−0.346 + 0.346i)13-s + (0.00584 − 0.489i)14-s − 0.126·15-s + (0.998 + 0.0477i)16-s − 1.38·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.904 + 0.426i$
Analytic conductor: \(4.99816\)
Root analytic conductor: \(2.23565\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :7/2),\ 0.904 + 0.426i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.94730 - 0.659819i\)
\(L(\frac12)\) \(\approx\) \(2.94730 - 0.659819i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-11.3 - 0.135i)T \)
good3 \( 1 + (-26.1 + 26.1i)T - 2.18e3iT^{2} \)
5 \( 1 + (31.6 + 31.6i)T + 7.81e4iT^{2} \)
7 \( 1 + 444. iT - 8.23e5T^{2} \)
11 \( 1 + (-472. - 472. i)T + 1.94e7iT^{2} \)
13 \( 1 + (2.74e3 - 2.74e3i)T - 6.27e7iT^{2} \)
17 \( 1 + 2.79e4T + 4.10e8T^{2} \)
19 \( 1 + (3.57e4 - 3.57e4i)T - 8.93e8iT^{2} \)
23 \( 1 - 1.89e4iT - 3.40e9T^{2} \)
29 \( 1 + (-1.01e5 + 1.01e5i)T - 1.72e10iT^{2} \)
31 \( 1 - 2.12e5T + 2.75e10T^{2} \)
37 \( 1 + (3.06e5 + 3.06e5i)T + 9.49e10iT^{2} \)
41 \( 1 - 1.52e5iT - 1.94e11T^{2} \)
43 \( 1 + (3.51e3 + 3.51e3i)T + 2.71e11iT^{2} \)
47 \( 1 - 5.30e5T + 5.06e11T^{2} \)
53 \( 1 + (1.04e6 + 1.04e6i)T + 1.17e12iT^{2} \)
59 \( 1 + (1.13e6 + 1.13e6i)T + 2.48e12iT^{2} \)
61 \( 1 + (-1.95e5 + 1.95e5i)T - 3.14e12iT^{2} \)
67 \( 1 + (1.52e6 - 1.52e6i)T - 6.06e12iT^{2} \)
71 \( 1 + 4.95e6iT - 9.09e12T^{2} \)
73 \( 1 - 5.65e6iT - 1.10e13T^{2} \)
79 \( 1 - 5.88e6T + 1.92e13T^{2} \)
83 \( 1 + (1.38e6 - 1.38e6i)T - 2.71e13iT^{2} \)
89 \( 1 + 2.76e6iT - 4.42e13T^{2} \)
97 \( 1 - 4.24e6T + 8.07e13T^{2} \)
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

−17.19173199218317282510457440021, −15.81267189764415568439969737625, −14.35150579216875038686281339339, −13.46016340401633083384563133857, −12.23774517764295380004140998142, −10.58970475004342795600416874417, −8.118781358288492901138633453367, −6.62207202059194736113272432704, −4.36039301134535876329581533167, −2.16040250418056141149621549132, 2.79105941828633806011252030319, 4.55593171561648294461514532515, 6.61550578772967177443281877804, 8.821070917695681058409393790262, 10.70753419401637342094324101626, 12.22687300965609890975371229743, 13.64217788465693675144860310416, 15.11611633756207793875694379194, 15.52819399208359090047372671185, 17.37373843845715330000267901996

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