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Properties

Label 2-2e4-16.3-c8-0-9
Degree $2$
Conductor $16$
Sign $0.958 + 0.285i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (15.6 − 3.52i)2-s + (−28.7 + 28.7i)3-s + (231. − 109. i)4-s + (301. − 301. i)5-s + (−347. + 550. i)6-s + 3.82e3·7-s + (3.22e3 − 2.53e3i)8-s + 4.90e3i·9-s + (3.64e3 − 5.76e3i)10-s + (−1.45e4 − 1.45e4i)11-s + (−3.48e3 + 9.82e3i)12-s + (1.20e4 + 1.20e4i)13-s + (5.97e4 − 1.34e4i)14-s + 1.73e4i·15-s + (4.13e4 − 5.08e4i)16-s − 7.60e4·17-s + ⋯
L(s)  = 1  + (0.975 − 0.220i)2-s + (−0.355 + 0.355i)3-s + (0.902 − 0.429i)4-s + (0.482 − 0.482i)5-s + (−0.268 + 0.425i)6-s + 1.59·7-s + (0.786 − 0.617i)8-s + 0.747i·9-s + (0.364 − 0.576i)10-s + (−0.993 − 0.993i)11-s + (−0.168 + 0.473i)12-s + (0.423 + 0.423i)13-s + (1.55 − 0.351i)14-s + 0.342i·15-s + (0.630 − 0.775i)16-s − 0.910·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.958 + 0.285i$
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: \((2,\ 16,\ (\ :4),\ 0.958 + 0.285i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.89932 - 0.423153i\)
\(L(\frac12)\) \(\approx\) \(2.89932 - 0.423153i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-15.6 + 3.52i)T \)
good3 \( 1 + (28.7 - 28.7i)T - 6.56e3iT^{2} \)
5 \( 1 + (-301. + 301. i)T - 3.90e5iT^{2} \)
7 \( 1 - 3.82e3T + 5.76e6T^{2} \)
11 \( 1 + (1.45e4 + 1.45e4i)T + 2.14e8iT^{2} \)
13 \( 1 + (-1.20e4 - 1.20e4i)T + 8.15e8iT^{2} \)
17 \( 1 + 7.60e4T + 6.97e9T^{2} \)
19 \( 1 + (6.63e4 - 6.63e4i)T - 1.69e10iT^{2} \)
23 \( 1 + 3.04e5T + 7.83e10T^{2} \)
29 \( 1 + (-3.11e5 - 3.11e5i)T + 5.00e11iT^{2} \)
31 \( 1 - 1.15e5iT - 8.52e11T^{2} \)
37 \( 1 + (-1.49e6 + 1.49e6i)T - 3.51e12iT^{2} \)
41 \( 1 - 3.73e6iT - 7.98e12T^{2} \)
43 \( 1 + (2.78e6 + 2.78e6i)T + 1.16e13iT^{2} \)
47 \( 1 + 4.37e6iT - 2.38e13T^{2} \)
53 \( 1 + (7.77e6 - 7.77e6i)T - 6.22e13iT^{2} \)
59 \( 1 + (-3.02e6 - 3.02e6i)T + 1.46e14iT^{2} \)
61 \( 1 + (9.87e6 + 9.87e6i)T + 1.91e14iT^{2} \)
67 \( 1 + (-5.11e6 + 5.11e6i)T - 4.06e14iT^{2} \)
71 \( 1 + 8.45e6T + 6.45e14T^{2} \)
73 \( 1 - 4.84e7iT - 8.06e14T^{2} \)
79 \( 1 - 6.19e7iT - 1.51e15T^{2} \)
83 \( 1 + (-5.28e7 + 5.28e7i)T - 2.25e15iT^{2} \)
89 \( 1 + 7.90e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.39e8T + 7.83e15T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.86061046667466329425078919544, −15.75345191571783219097472471021, −14.20481022029851604141870672804, −13.25338956017982407471609833900, −11.43609362430215701577498290741, −10.62287407247420106264306118283, −8.148189619771418840273679394617, −5.67259757279473208161836988997, −4.58902942421216412487634319468, −1.87943283885634044258563484132, 2.11868166038360489000330867076, 4.69695859669193376616048401110, 6.32317893522909171676378498243, 7.896579236236813287849846311208, 10.65332892236609240313536913098, 11.87891049596983376995228617544, 13.25361306531311758462970380373, 14.59728353815677506580771654441, 15.50045771097759530871695715478, 17.63852200307399363432249093562

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