↑

Properties

Label 2-2e4-16.13-c13-0-16
Degree $2$
Conductor $16$
Sign $0.108 + 0.994i$
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  + (28.6 + 85.8i)2-s + (−336. − 336. i)3-s + (−6.55e3 + 4.91e3i)4-s + (−6.82e3 + 6.82e3i)5-s + (1.92e4 − 3.85e4i)6-s + 5.09e5i·7-s + (−6.09e5 − 4.21e5i)8-s − 1.36e6i·9-s + (−7.81e5 − 3.90e5i)10-s + (1.18e6 − 1.18e6i)11-s + (3.86e6 + 5.51e5i)12-s + (−2.08e7 − 2.08e7i)13-s + (−4.37e7 + 1.45e7i)14-s + 4.59e6·15-s + (1.87e7 − 6.44e7i)16-s + 6.68e7·17-s + ⋯
L(s)  = 1  + (0.316 + 0.948i)2-s + (−0.266 − 0.266i)3-s + (−0.799 + 0.600i)4-s + (−0.195 + 0.195i)5-s + (0.168 − 0.337i)6-s + 1.63i·7-s + (−0.822 − 0.569i)8-s − 0.857i·9-s + (−0.246 − 0.123i)10-s + (0.201 − 0.201i)11-s + (0.373 + 0.0533i)12-s + (−1.19 − 1.19i)13-s + (−1.55 + 0.517i)14-s + 0.104·15-s + (0.279 − 0.960i)16-s + 0.672·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.108 + 0.994i$
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: \((2,\ 16,\ (\ :13/2),\ 0.108 + 0.994i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.202410 - 0.181459i\)
\(L(\frac12)\) \(\approx\) \(0.202410 - 0.181459i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-28.6 - 85.8i)T \)
good3 \( 1 + (336. + 336. i)T + 1.59e6iT^{2} \)
5 \( 1 + (6.82e3 - 6.82e3i)T - 1.22e9iT^{2} \)
7 \( 1 - 5.09e5iT - 9.68e10T^{2} \)
11 \( 1 + (-1.18e6 + 1.18e6i)T - 3.45e13iT^{2} \)
13 \( 1 + (2.08e7 + 2.08e7i)T + 3.02e14iT^{2} \)
17 \( 1 - 6.68e7T + 9.90e15T^{2} \)
19 \( 1 + (6.18e6 + 6.18e6i)T + 4.20e16iT^{2} \)
23 \( 1 + 1.35e9iT - 5.04e17T^{2} \)
29 \( 1 + (2.99e9 + 2.99e9i)T + 1.02e19iT^{2} \)
31 \( 1 - 8.37e8T + 2.44e19T^{2} \)
37 \( 1 + (1.19e10 - 1.19e10i)T - 2.43e20iT^{2} \)
41 \( 1 - 1.93e10iT - 9.25e20T^{2} \)
43 \( 1 + (8.03e9 - 8.03e9i)T - 1.71e21iT^{2} \)
47 \( 1 + 9.68e10T + 5.46e21T^{2} \)
53 \( 1 + (2.77e9 - 2.77e9i)T - 2.60e22iT^{2} \)
59 \( 1 + (4.89e10 - 4.89e10i)T - 1.04e23iT^{2} \)
61 \( 1 + (3.75e11 + 3.75e11i)T + 1.61e23iT^{2} \)
67 \( 1 + (2.29e11 + 2.29e11i)T + 5.48e23iT^{2} \)
71 \( 1 + 1.12e12iT - 1.16e24T^{2} \)
73 \( 1 - 1.43e12iT - 1.67e24T^{2} \)
79 \( 1 - 2.60e12T + 4.66e24T^{2} \)
83 \( 1 + (-2.17e12 - 2.17e12i)T + 8.87e24iT^{2} \)
89 \( 1 - 4.02e11iT - 2.19e25T^{2} \)
97 \( 1 + 4.42e12T + 6.73e25T^{2} \)
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

−15.23092827881049138829868603384, −14.80919873049031165469274485797, −12.71847731670589405385758776336, −11.99327179576741477788542296712, −9.473111336693985816999006976807, −8.070199568988243511806247840809, −6.40026710171808380873888752120, −5.23956248722264235214718972550, −3.02110582973013095053761569034, −0.096901920067049292006271964037, 1.63809592107297870886882321951, 3.84769395500937253588414416593, 4.98280361827148780726830278295, 7.42149410055256834324526321233, 9.628851879321381892639545566192, 10.69398407009130575485686794421, 11.92438600652321850129235830521, 13.49299155874572466897166691074, 14.39947199039659606543189343515, 16.45784620387029510685793191966

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