↑

Properties

Label 2-2e4-16.3-c8-0-2
Degree $2$
Conductor $16$
Sign $-0.0623 + 0.998i$
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  + (−1.81 + 15.8i)2-s + (−71.2 + 71.2i)3-s + (−249. − 57.7i)4-s + (−8.38 + 8.38i)5-s + (−1.00e3 − 1.26e3i)6-s + 1.08e3·7-s + (1.37e3 − 3.85e3i)8-s − 3.59e3i·9-s + (−118. − 148. i)10-s + (983. + 983. i)11-s + (2.18e4 − 1.36e4i)12-s + (−3.82e4 − 3.82e4i)13-s + (−1.97e3 + 1.72e4i)14-s − 1.19e3i·15-s + (5.88e4 + 2.88e4i)16-s − 6.81e4·17-s + ⋯
L(s)  = 1  + (−0.113 + 0.993i)2-s + (−0.879 + 0.879i)3-s + (−0.974 − 0.225i)4-s + (−0.0134 + 0.0134i)5-s + (−0.774 − 0.973i)6-s + 0.452·7-s + (0.334 − 0.942i)8-s − 0.547i·9-s + (−0.0118 − 0.0148i)10-s + (0.0671 + 0.0671i)11-s + (1.05 − 0.658i)12-s + (−1.33 − 1.33i)13-s + (−0.0514 + 0.450i)14-s − 0.0236i·15-s + (0.898 + 0.439i)16-s − 0.816·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0623 + 0.998i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.0623 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.0623 + 0.998i$
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: \((2,\ 16,\ (\ :4),\ -0.0623 + 0.998i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.104305 - 0.111026i\)
\(L(\frac12)\) \(\approx\) \(0.104305 - 0.111026i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.81 - 15.8i)T \)
good3 \( 1 + (71.2 - 71.2i)T - 6.56e3iT^{2} \)
5 \( 1 + (8.38 - 8.38i)T - 3.90e5iT^{2} \)
7 \( 1 - 1.08e3T + 5.76e6T^{2} \)
11 \( 1 + (-983. - 983. i)T + 2.14e8iT^{2} \)
13 \( 1 + (3.82e4 + 3.82e4i)T + 8.15e8iT^{2} \)
17 \( 1 + 6.81e4T + 6.97e9T^{2} \)
19 \( 1 + (1.16e5 - 1.16e5i)T - 1.69e10iT^{2} \)
23 \( 1 - 2.17e5T + 7.83e10T^{2} \)
29 \( 1 + (7.27e5 + 7.27e5i)T + 5.00e11iT^{2} \)
31 \( 1 + 5.97e5iT - 8.52e11T^{2} \)
37 \( 1 + (5.30e5 - 5.30e5i)T - 3.51e12iT^{2} \)
41 \( 1 + 2.04e6iT - 7.98e12T^{2} \)
43 \( 1 + (-3.58e6 - 3.58e6i)T + 1.16e13iT^{2} \)
47 \( 1 - 5.76e6iT - 2.38e13T^{2} \)
53 \( 1 + (6.59e6 - 6.59e6i)T - 6.22e13iT^{2} \)
59 \( 1 + (1.24e7 + 1.24e7i)T + 1.46e14iT^{2} \)
61 \( 1 + (-1.17e7 - 1.17e7i)T + 1.91e14iT^{2} \)
67 \( 1 + (1.15e7 - 1.15e7i)T - 4.06e14iT^{2} \)
71 \( 1 + 3.83e5T + 6.45e14T^{2} \)
73 \( 1 + 1.19e7iT - 8.06e14T^{2} \)
79 \( 1 - 1.63e7iT - 1.51e15T^{2} \)
83 \( 1 + (5.74e7 - 5.74e7i)T - 2.25e15iT^{2} \)
89 \( 1 + 3.87e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.09e8T + 7.83e15T^{2} \)
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

−17.43469567879580057018867895265, −17.04904185827189963371974163254, −15.56062143367860306690572376654, −14.76056421076874801558286671129, −12.85190164358164707951925157660, −10.90998366376239392736304839154, −9.630710564285805245944352230877, −7.73513368312555462539173476174, −5.74332029384802211026566890554, −4.53802214624413673978020130815, 0.096682504930645028345452262032, 1.95102417621431996538629407737, 4.76977510409019747690189712114, 6.96972308346025390627551956527, 9.035278037146185322898650410086, 10.95048919859999034879058272744, 11.93023085623131089686939340674, 12.96909821534381153092500108543, 14.47908837812537078193754054885, 16.92255997291379515529094734563

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