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Properties

Label 2-2e4-16.3-c8-0-1
Degree $2$
Conductor $16$
Sign $-0.854 + 0.519i$
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  + (−5.55 + 15.0i)2-s + (65.0 − 65.0i)3-s + (−194. − 166. i)4-s + (−526. + 526. i)5-s + (615. + 1.33e3i)6-s − 3.63e3·7-s + (3.57e3 − 1.99e3i)8-s − 1.90e3i·9-s + (−4.98e3 − 1.08e4i)10-s + (−1.99e4 − 1.99e4i)11-s + (−2.34e4 + 1.80e3i)12-s + (2.04e4 + 2.04e4i)13-s + (2.02e4 − 5.46e4i)14-s + 6.85e4i·15-s + (1.00e4 + 6.47e4i)16-s − 5.46e4·17-s + ⋯
L(s)  = 1  + (−0.346 + 0.937i)2-s + (0.803 − 0.803i)3-s + (−0.759 − 0.650i)4-s + (−0.842 + 0.842i)5-s + (0.474 + 1.03i)6-s − 1.51·7-s + (0.873 − 0.486i)8-s − 0.290i·9-s + (−0.498 − 1.08i)10-s + (−1.36 − 1.36i)11-s + (−1.13 + 0.0871i)12-s + (0.717 + 0.717i)13-s + (0.525 − 1.42i)14-s + 1.35i·15-s + (0.152 + 0.988i)16-s − 0.654·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 + 0.519i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.854 + 0.519i$
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: \((2,\ 16,\ (\ :4),\ -0.854 + 0.519i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0107289 - 0.0383040i\)
\(L(\frac12)\) \(\approx\) \(0.0107289 - 0.0383040i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.55 - 15.0i)T \)
good3 \( 1 + (-65.0 + 65.0i)T - 6.56e3iT^{2} \)
5 \( 1 + (526. - 526. i)T - 3.90e5iT^{2} \)
7 \( 1 + 3.63e3T + 5.76e6T^{2} \)
11 \( 1 + (1.99e4 + 1.99e4i)T + 2.14e8iT^{2} \)
13 \( 1 + (-2.04e4 - 2.04e4i)T + 8.15e8iT^{2} \)
17 \( 1 + 5.46e4T + 6.97e9T^{2} \)
19 \( 1 + (-2.59e4 + 2.59e4i)T - 1.69e10iT^{2} \)
23 \( 1 + 8.51e4T + 7.83e10T^{2} \)
29 \( 1 + (-1.29e5 - 1.29e5i)T + 5.00e11iT^{2} \)
31 \( 1 - 1.57e5iT - 8.52e11T^{2} \)
37 \( 1 + (-2.19e4 + 2.19e4i)T - 3.51e12iT^{2} \)
41 \( 1 + 8.02e5iT - 7.98e12T^{2} \)
43 \( 1 + (3.57e5 + 3.57e5i)T + 1.16e13iT^{2} \)
47 \( 1 + 4.64e6iT - 2.38e13T^{2} \)
53 \( 1 + (9.82e6 - 9.82e6i)T - 6.22e13iT^{2} \)
59 \( 1 + (6.70e6 + 6.70e6i)T + 1.46e14iT^{2} \)
61 \( 1 + (1.18e7 + 1.18e7i)T + 1.91e14iT^{2} \)
67 \( 1 + (2.29e7 - 2.29e7i)T - 4.06e14iT^{2} \)
71 \( 1 + 1.38e7T + 6.45e14T^{2} \)
73 \( 1 + 2.83e7iT - 8.06e14T^{2} \)
79 \( 1 - 5.72e6iT - 1.51e15T^{2} \)
83 \( 1 + (8.26e6 - 8.26e6i)T - 2.25e15iT^{2} \)
89 \( 1 + 8.44e7iT - 3.93e15T^{2} \)
97 \( 1 - 4.87e7T + 7.83e15T^{2} \)
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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

−18.54662619232036931742195132649, −16.25578858819094515565625370270, −15.58500117797788959342979415215, −13.90527085724840668852697647199, −13.16254283722812982582587763710, −10.72281704670212246635427846625, −8.797181286534376058445825980206, −7.54895030320972442362967495439, −6.36353368312376669501796505069, −3.20187975695602888397328063282, 0.02184236191181538098370023134, 3.02020397287353269694278164425, 4.38613651949249161013325408929, 8.025328349612448121601574202821, 9.345697159852054574040618725038, 10.32481285768482047302505677814, 12.40313988144087720438801064655, 13.20730611229755919140658762965, 15.45964421398610077888833668319, 16.15017902923322465819109215526

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