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Properties

Label 2-2e4-16.3-c12-0-8
Degree $2$
Conductor $16$
Sign $0.573 - 0.818i$
Analytic cond. $14.6239$
Root an. cond. $3.82412$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−55.0 + 32.5i)2-s + (769. − 769. i)3-s + (1.97e3 − 3.59e3i)4-s + (−1.76e4 + 1.76e4i)5-s + (−1.73e4 + 6.75e4i)6-s + 1.53e4·7-s + (8.49e3 + 2.62e5i)8-s − 6.54e5i·9-s + (3.96e5 − 1.54e6i)10-s + (2.33e5 + 2.33e5i)11-s + (−1.24e6 − 4.28e6i)12-s + (3.01e6 + 3.01e6i)13-s + (−8.47e5 + 5.01e5i)14-s + 2.71e7i·15-s + (−9.00e6 − 1.41e7i)16-s + 4.44e7·17-s + ⋯
L(s)  = 1  + (−0.860 + 0.509i)2-s + (1.05 − 1.05i)3-s + (0.481 − 0.876i)4-s + (−1.12 + 1.12i)5-s + (−0.370 + 1.44i)6-s + 0.130·7-s + (0.0323 + 0.999i)8-s − 1.23i·9-s + (0.396 − 1.54i)10-s + (0.131 + 0.131i)11-s + (−0.417 − 1.43i)12-s + (0.624 + 0.624i)13-s + (−0.112 + 0.0666i)14-s + 2.38i·15-s + (−0.536 − 0.843i)16-s + 1.84·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.818i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.573 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.573 - 0.818i$
Analytic conductor: \(14.6239\)
Root analytic conductor: \(3.82412\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :6),\ 0.573 - 0.818i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(1.23810 + 0.644174i\)
\(L(\frac12)\) \(\approx\) \(1.23810 + 0.644174i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (55.0 - 32.5i)T \)
good3 \( 1 + (-769. + 769. i)T - 5.31e5iT^{2} \)
5 \( 1 + (1.76e4 - 1.76e4i)T - 2.44e8iT^{2} \)
7 \( 1 - 1.53e4T + 1.38e10T^{2} \)
11 \( 1 + (-2.33e5 - 2.33e5i)T + 3.13e12iT^{2} \)
13 \( 1 + (-3.01e6 - 3.01e6i)T + 2.32e13iT^{2} \)
17 \( 1 - 4.44e7T + 5.82e14T^{2} \)
19 \( 1 + (5.03e7 - 5.03e7i)T - 2.21e15iT^{2} \)
23 \( 1 - 1.18e8T + 2.19e16T^{2} \)
29 \( 1 + (-6.55e8 - 6.55e8i)T + 3.53e17iT^{2} \)
31 \( 1 - 9.22e8iT - 7.87e17T^{2} \)
37 \( 1 + (-6.00e8 + 6.00e8i)T - 6.58e18iT^{2} \)
41 \( 1 + 1.69e9iT - 2.25e19T^{2} \)
43 \( 1 + (3.66e9 + 3.66e9i)T + 3.99e19iT^{2} \)
47 \( 1 - 1.00e10iT - 1.16e20T^{2} \)
53 \( 1 + (-2.00e9 + 2.00e9i)T - 4.91e20iT^{2} \)
59 \( 1 + (6.20e8 + 6.20e8i)T + 1.77e21iT^{2} \)
61 \( 1 + (-3.14e10 - 3.14e10i)T + 2.65e21iT^{2} \)
67 \( 1 + (-8.76e9 + 8.76e9i)T - 8.18e21iT^{2} \)
71 \( 1 + 2.33e11T + 1.64e22T^{2} \)
73 \( 1 + 2.06e11iT - 2.29e22T^{2} \)
79 \( 1 + 1.90e11iT - 5.90e22T^{2} \)
83 \( 1 + (1.37e11 - 1.37e11i)T - 1.06e23iT^{2} \)
89 \( 1 - 7.54e11iT - 2.46e23T^{2} \)
97 \( 1 - 2.77e11T + 6.93e23T^{2} \)
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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

−16.30583533937127654453116605669, −14.75390696488273470700152883126, −14.29382936498467514255094565824, −12.12686407595991937982515622631, −10.56850471235673524698696142592, −8.565209035667252057869813916750, −7.62479448704899757173774978058, −6.65193848709319849614360788074, −3.17775829485864685332396314506, −1.40634554246283957317685683977, 0.75838063628690549148551892318, 3.15180128697104211517487296451, 4.36678801792194121790838197308, 8.013049952282612377373865964546, 8.657617514777940587940472625246, 9.956240890899665652548977496496, 11.51764247291581791806495575225, 12.98405205262046282415589241100, 15.09370431349456978055852208296, 15.98921778045040888906557651929

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