↑

Properties

Label 2-2e4-16.13-c21-0-40
Degree $2$
Conductor $16$
Sign $0.744 - 0.667i$
Analytic cond. $44.7163$
Root an. cond. $6.68703$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (933. − 1.10e3i)2-s + (−9.97e4 − 9.97e4i)3-s + (−3.52e5 − 2.06e6i)4-s + (1.64e7 − 1.64e7i)5-s + (−2.03e8 + 1.72e7i)6-s − 1.30e7i·7-s + (−2.61e9 − 1.54e9i)8-s + 9.43e9i·9-s + (−2.84e9 − 3.35e10i)10-s + (−7.19e10 + 7.19e10i)11-s + (−1.71e11 + 2.41e11i)12-s + (3.46e11 + 3.46e11i)13-s + (−1.44e10 − 1.21e10i)14-s − 3.27e12·15-s + (−4.14e12 + 1.45e12i)16-s − 4.87e12·17-s + ⋯
L(s)  = 1  + (0.644 − 0.764i)2-s + (−0.975 − 0.975i)3-s + (−0.168 − 0.985i)4-s + (0.752 − 0.752i)5-s + (−1.37 + 0.116i)6-s − 0.0174i·7-s + (−0.861 − 0.507i)8-s + 0.902i·9-s + (−0.0898 − 1.06i)10-s + (−0.836 + 0.836i)11-s + (−0.797 + 1.12i)12-s + (0.697 + 0.697i)13-s + (−0.0133 − 0.0112i)14-s − 1.46·15-s + (−0.943 + 0.331i)16-s − 0.586·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.744 - 0.667i$
Analytic conductor: \(44.7163\)
Root analytic conductor: \(6.68703\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :21/2),\ 0.744 - 0.667i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.1382402005\)
\(L(\frac12)\) \(\approx\) \(0.1382402005\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-933. + 1.10e3i)T \)
good3 \( 1 + (9.97e4 + 9.97e4i)T + 1.04e10iT^{2} \)
5 \( 1 + (-1.64e7 + 1.64e7i)T - 4.76e14iT^{2} \)
7 \( 1 + 1.30e7iT - 5.58e17T^{2} \)
11 \( 1 + (7.19e10 - 7.19e10i)T - 7.40e21iT^{2} \)
13 \( 1 + (-3.46e11 - 3.46e11i)T + 2.47e23iT^{2} \)
17 \( 1 + 4.87e12T + 6.90e25T^{2} \)
19 \( 1 + (2.92e13 + 2.92e13i)T + 7.14e26iT^{2} \)
23 \( 1 - 1.46e14iT - 3.94e28T^{2} \)
29 \( 1 + (2.05e13 + 2.05e13i)T + 5.13e30iT^{2} \)
31 \( 1 - 4.09e15T + 2.08e31T^{2} \)
37 \( 1 + (8.05e15 - 8.05e15i)T - 8.55e32iT^{2} \)
41 \( 1 + 1.29e16iT - 7.38e33T^{2} \)
43 \( 1 + (-1.82e17 + 1.82e17i)T - 2.00e34iT^{2} \)
47 \( 1 + 5.14e17T + 1.30e35T^{2} \)
53 \( 1 + (-2.86e17 + 2.86e17i)T - 1.62e36iT^{2} \)
59 \( 1 + (3.00e18 - 3.00e18i)T - 1.54e37iT^{2} \)
61 \( 1 + (-6.81e16 - 6.81e16i)T + 3.10e37iT^{2} \)
67 \( 1 + (7.96e18 + 7.96e18i)T + 2.22e38iT^{2} \)
71 \( 1 - 2.62e19iT - 7.52e38T^{2} \)
73 \( 1 + 7.20e19iT - 1.34e39T^{2} \)
79 \( 1 + 1.58e20T + 7.08e39T^{2} \)
83 \( 1 + (1.06e20 + 1.06e20i)T + 1.99e40iT^{2} \)
89 \( 1 - 4.00e20iT - 8.65e40T^{2} \)
97 \( 1 + 6.61e20T + 5.27e41T^{2} \)
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

−12.98902932156835561164314654919, −11.88518510261811413839274593073, −10.68159508436482320538484316577, −9.112225707243875400980060856738, −6.80138015528682701087585399867, −5.66908426921937827082076791070, −4.55789643941542359545328014164, −2.20917566732200416216463692993, −1.29664379618535546924140083032, −0.03443208431290463958874069546, 2.78953292068554231273959482565, 4.28988015706796860437752367858, 5.68989966516412319575577509108, 6.29350409550034093114308894394, 8.312059971079090919112356774660, 10.25081959390321381257827242841, 11.12048998798727979887256551299, 12.88966658916404997409964393441, 14.18498331180781256176593138588, 15.46559381048498410232907686843

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