↑

Properties

Label 2-2e4-16.13-c27-0-27
Degree $2$
Conductor $16$
Sign $0.971 + 0.235i$
Analytic cond. $73.8968$
Root an. cond. $8.59633$
Motivic weight $27$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.14e4 − 1.81e3i)2-s + (1.53e5 + 1.53e5i)3-s + (1.27e8 + 4.15e7i)4-s + (2.87e9 − 2.87e9i)5-s + (−1.47e9 − 2.03e9i)6-s + 4.55e11i·7-s + (−1.38e12 − 7.07e11i)8-s − 7.57e12i·9-s + (−3.80e13 + 2.76e13i)10-s + (2.18e13 − 2.18e13i)11-s + (1.32e13 + 2.60e13i)12-s + (6.58e13 + 6.58e13i)13-s + (8.27e14 − 5.20e15i)14-s + 8.82e14·15-s + (1.45e16 + 1.06e16i)16-s + 3.33e16·17-s + ⋯
L(s)  = 1  + (−0.987 − 0.156i)2-s + (0.0556 + 0.0556i)3-s + (0.950 + 0.309i)4-s + (1.05 − 1.05i)5-s + (−0.0462 − 0.0636i)6-s + 1.77i·7-s + (−0.890 − 0.455i)8-s − 0.993i·9-s + (−1.20 + 0.873i)10-s + (0.190 − 0.190i)11-s + (0.0356 + 0.0701i)12-s + (0.0603 + 0.0603i)13-s + (0.278 − 1.75i)14-s + 0.117·15-s + (0.808 + 0.589i)16-s + 0.816·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.971 + 0.235i$
Analytic conductor: \(73.8968\)
Root analytic conductor: \(8.59633\)
Motivic weight: \(27\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :27/2),\ 0.971 + 0.235i)\)

Particular Values

\(L(14)\) \(\approx\) \(1.909271770\)
\(L(\frac12)\) \(\approx\) \(1.909271770\)
\(L(\frac{29}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.14e4 + 1.81e3i)T \)
good3 \( 1 + (-1.53e5 - 1.53e5i)T + 7.62e12iT^{2} \)
5 \( 1 + (-2.87e9 + 2.87e9i)T - 7.45e18iT^{2} \)
7 \( 1 - 4.55e11iT - 6.57e22T^{2} \)
11 \( 1 + (-2.18e13 + 2.18e13i)T - 1.31e28iT^{2} \)
13 \( 1 + (-6.58e13 - 6.58e13i)T + 1.19e30iT^{2} \)
17 \( 1 - 3.33e16T + 1.66e33T^{2} \)
19 \( 1 + (-2.22e17 - 2.22e17i)T + 3.36e34iT^{2} \)
23 \( 1 + 4.14e18iT - 5.84e36T^{2} \)
29 \( 1 + (-1.87e19 - 1.87e19i)T + 3.05e39iT^{2} \)
31 \( 1 + 5.35e19T + 1.84e40T^{2} \)
37 \( 1 + (1.64e21 - 1.64e21i)T - 2.19e42iT^{2} \)
41 \( 1 - 3.79e21iT - 3.50e43T^{2} \)
43 \( 1 + (-4.06e21 + 4.06e21i)T - 1.26e44iT^{2} \)
47 \( 1 - 3.33e22T + 1.40e45T^{2} \)
53 \( 1 + (7.44e22 - 7.44e22i)T - 3.59e46iT^{2} \)
59 \( 1 + (-3.85e23 + 3.85e23i)T - 6.50e47iT^{2} \)
61 \( 1 + (-3.77e23 - 3.77e23i)T + 1.59e48iT^{2} \)
67 \( 1 + (-1.29e24 - 1.29e24i)T + 2.01e49iT^{2} \)
71 \( 1 + 1.59e25iT - 9.63e49T^{2} \)
73 \( 1 - 1.45e25iT - 2.04e50T^{2} \)
79 \( 1 - 3.96e25T + 1.72e51T^{2} \)
83 \( 1 + (2.74e25 + 2.74e25i)T + 6.53e51iT^{2} \)
89 \( 1 - 1.01e26iT - 4.30e52T^{2} \)
97 \( 1 - 5.84e26T + 4.39e53T^{2} \)
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

−12.44283407546434527907263664617, −12.04196940716129625202403193662, −9.955923661993306853621185831382, −9.117982495180216329296650984198, −8.406932819724664413845543673308, −6.27673517944028928233801501935, −5.44110128094974632395916508806, −3.09353767280761871271290337562, −1.82755633013662156051619433355, −0.865206185336428637119140300180, 0.850575904183178792192576853660, 1.94876025603849599209867784113, 3.29760853300294552866258131228, 5.48347884687698229228726370071, 7.06165303233027603794112528232, 7.51259314933724282069589425105, 9.575790436158647142431013915554, 10.39426666181940888494137017447, 11.19501263564207131832419086566, 13.60080845158248424448084788127

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