↑

Properties

Label 2-2e4-16.13-c5-0-4
Degree $2$
Conductor $16$
Sign $0.535 + 0.844i$
Analytic cond. $2.56614$
Root an. cond. $1.60191$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−5.12 + 2.38i)2-s + (−4.57 − 4.57i)3-s + (20.5 − 24.4i)4-s + (48.5 − 48.5i)5-s + (34.4 + 12.5i)6-s − 106. i·7-s + (−47.1 + 174. i)8-s − 201. i·9-s + (−133. + 365. i)10-s + (−184. + 184. i)11-s + (−206. + 17.8i)12-s + (3.64 + 3.64i)13-s + (253. + 545. i)14-s − 445.·15-s + (−175. − 1.00e3i)16-s + 2.06e3·17-s + ⋯
L(s)  = 1  + (−0.906 + 0.422i)2-s + (−0.293 − 0.293i)3-s + (0.643 − 0.765i)4-s + (0.869 − 0.869i)5-s + (0.390 + 0.142i)6-s − 0.820i·7-s + (−0.260 + 0.965i)8-s − 0.827i·9-s + (−0.421 + 1.15i)10-s + (−0.460 + 0.460i)11-s + (−0.413 + 0.0357i)12-s + (0.00598 + 0.00598i)13-s + (0.346 + 0.743i)14-s − 0.510·15-s + (−0.171 − 0.985i)16-s + 1.73·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.535 + 0.844i$
Analytic conductor: \(2.56614\)
Root analytic conductor: \(1.60191\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :5/2),\ 0.535 + 0.844i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.768368 - 0.422547i\)
\(L(\frac12)\) \(\approx\) \(0.768368 - 0.422547i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.12 - 2.38i)T \)
good3 \( 1 + (4.57 + 4.57i)T + 243iT^{2} \)
5 \( 1 + (-48.5 + 48.5i)T - 3.12e3iT^{2} \)
7 \( 1 + 106. iT - 1.68e4T^{2} \)
11 \( 1 + (184. - 184. i)T - 1.61e5iT^{2} \)
13 \( 1 + (-3.64 - 3.64i)T + 3.71e5iT^{2} \)
17 \( 1 - 2.06e3T + 1.41e6T^{2} \)
19 \( 1 + (1.25e3 + 1.25e3i)T + 2.47e6iT^{2} \)
23 \( 1 - 3.86e3iT - 6.43e6T^{2} \)
29 \( 1 + (-1.95e3 - 1.95e3i)T + 2.05e7iT^{2} \)
31 \( 1 - 2.76e3T + 2.86e7T^{2} \)
37 \( 1 + (6.43e3 - 6.43e3i)T - 6.93e7iT^{2} \)
41 \( 1 + 1.12e4iT - 1.15e8T^{2} \)
43 \( 1 + (-1.29e4 + 1.29e4i)T - 1.47e8iT^{2} \)
47 \( 1 - 2.58e4T + 2.29e8T^{2} \)
53 \( 1 + (2.18e4 - 2.18e4i)T - 4.18e8iT^{2} \)
59 \( 1 + (1.41e4 - 1.41e4i)T - 7.14e8iT^{2} \)
61 \( 1 + (-6.21e3 - 6.21e3i)T + 8.44e8iT^{2} \)
67 \( 1 + (-2.86e4 - 2.86e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 2.94e4iT - 1.80e9T^{2} \)
73 \( 1 + 2.85e3iT - 2.07e9T^{2} \)
79 \( 1 + 2.33e4T + 3.07e9T^{2} \)
83 \( 1 + (2.45e4 + 2.45e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 1.93e4iT - 5.58e9T^{2} \)
97 \( 1 - 4.42e4T + 8.58e9T^{2} \)
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

−17.35656881777269977078863792175, −17.18631404150900340797035644763, −15.52991144251810431429232169281, −13.86135333281526816291850484076, −12.26596019237312068847664043413, −10.33938706200304417709440071187, −9.147953835585249685222075089018, −7.30571105947625047056530611398, −5.59594497865807997875697109447, −1.09421852096330593715596641919, 2.51787414634529933117206013263, 6.02647101830347376283519766824, 8.136792546909962444077176457892, 9.980013779479756289919712756231, 10.80625872088622125973855673090, 12.45423332434954833405648266789, 14.32898587675213525986629774709, 16.03560603413599035462573217757, 17.12855355687352692945941044104, 18.53616292838546271005849638061

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