Properties

Label 2-2e4-16.13-c13-0-6
Degree $2$
Conductor $16$
Sign $0.566 + 0.824i$
Analytic cond. $17.1569$
Root an. cond. $4.14209$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−81.6 + 38.9i)2-s + (−1.31e3 − 1.31e3i)3-s + (5.15e3 − 6.36e3i)4-s + (−4.38e4 + 4.38e4i)5-s + (1.58e5 + 5.62e4i)6-s + 1.56e5i·7-s + (−1.73e5 + 7.20e5i)8-s + 1.87e6i·9-s + (1.87e6 − 5.28e6i)10-s + (−8.14e6 + 8.14e6i)11-s + (−1.51e7 + 1.59e6i)12-s + (−1.01e7 − 1.01e7i)13-s + (−6.10e6 − 1.28e7i)14-s + 1.15e8·15-s + (−1.39e7 − 6.56e7i)16-s + 1.26e7·17-s + ⋯
L(s)  = 1  + (−0.902 + 0.430i)2-s + (−1.04 − 1.04i)3-s + (0.629 − 0.777i)4-s + (−1.25 + 1.25i)5-s + (1.38 + 0.492i)6-s + 0.503i·7-s + (−0.233 + 0.972i)8-s + 1.17i·9-s + (0.592 − 1.67i)10-s + (−1.38 + 1.38i)11-s + (−1.46 + 0.153i)12-s + (−0.581 − 0.581i)13-s + (−0.216 − 0.454i)14-s + 2.61·15-s + (−0.207 − 0.978i)16-s + 0.127·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 + 0.824i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.566 + 0.824i$
Analytic conductor: \(17.1569\)
Root analytic conductor: \(4.14209\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :13/2),\ 0.566 + 0.824i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.112938 - 0.0594438i\)
\(L(\frac12)\) \(\approx\) \(0.112938 - 0.0594438i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (81.6 - 38.9i)T \)
good3 \( 1 + (1.31e3 + 1.31e3i)T + 1.59e6iT^{2} \)
5 \( 1 + (4.38e4 - 4.38e4i)T - 1.22e9iT^{2} \)
7 \( 1 - 1.56e5iT - 9.68e10T^{2} \)
11 \( 1 + (8.14e6 - 8.14e6i)T - 3.45e13iT^{2} \)
13 \( 1 + (1.01e7 + 1.01e7i)T + 3.02e14iT^{2} \)
17 \( 1 - 1.26e7T + 9.90e15T^{2} \)
19 \( 1 + (-6.63e7 - 6.63e7i)T + 4.20e16iT^{2} \)
23 \( 1 - 2.44e8iT - 5.04e17T^{2} \)
29 \( 1 + (2.98e9 + 2.98e9i)T + 1.02e19iT^{2} \)
31 \( 1 + 6.35e9T + 2.44e19T^{2} \)
37 \( 1 + (1.76e9 - 1.76e9i)T - 2.43e20iT^{2} \)
41 \( 1 + 6.23e9iT - 9.25e20T^{2} \)
43 \( 1 + (-1.35e10 + 1.35e10i)T - 1.71e21iT^{2} \)
47 \( 1 - 5.50e10T + 5.46e21T^{2} \)
53 \( 1 + (6.19e10 - 6.19e10i)T - 2.60e22iT^{2} \)
59 \( 1 + (1.30e11 - 1.30e11i)T - 1.04e23iT^{2} \)
61 \( 1 + (-2.02e11 - 2.02e11i)T + 1.61e23iT^{2} \)
67 \( 1 + (-7.26e11 - 7.26e11i)T + 5.48e23iT^{2} \)
71 \( 1 + 4.02e11iT - 1.16e24T^{2} \)
73 \( 1 - 8.35e11iT - 1.67e24T^{2} \)
79 \( 1 + 1.82e12T + 4.66e24T^{2} \)
83 \( 1 + (2.72e10 + 2.72e10i)T + 8.87e24iT^{2} \)
89 \( 1 + 5.27e12iT - 2.19e25T^{2} \)
97 \( 1 - 1.27e13T + 6.73e25T^{2} \)
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

−15.70334720594606450535492419555, −14.92774540422760213195457862410, −12.47286684783673030762582660604, −11.42111527119750879698972705419, −10.28507532304305923120570653887, −7.59245524946835345028104597240, −7.28502300257299469747256255741, −5.59849034308187070319002635115, −2.32417534524250832563616241193, −0.14676924756081917018143490847, 0.57910472319364357306514383634, 3.72881423032734954583975347400, 5.15302886351821606424565715243, 7.71177148009050600406561171823, 9.088911410598848659522067183254, 10.71422373910801083118662179397, 11.47246923853068596863185076547, 12.76778257141693797928298617910, 15.72507209942902444971076453382, 16.35445161233954050067787951943

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