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Properties

Label 2-2e4-16.3-c12-0-10
Degree $2$
Conductor $16$
Sign $0.905 + 0.423i$
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  + (−11.7 − 62.9i)2-s + (252. − 252. i)3-s + (−3.81e3 + 1.48e3i)4-s + (−9.56e3 + 9.56e3i)5-s + (−1.88e4 − 1.29e4i)6-s + 1.68e5·7-s + (1.38e5 + 2.22e5i)8-s + 4.03e5i·9-s + (7.14e5 + 4.89e5i)10-s + (−1.62e6 − 1.62e6i)11-s + (−5.89e5 + 1.33e6i)12-s + (6.06e6 + 6.06e6i)13-s + (−1.98e6 − 1.06e7i)14-s + 4.83e6i·15-s + (1.23e7 − 1.13e7i)16-s − 7.47e6·17-s + ⋯
L(s)  = 1  + (−0.184 − 0.982i)2-s + (0.346 − 0.346i)3-s + (−0.932 + 0.361i)4-s + (−0.612 + 0.612i)5-s + (−0.404 − 0.276i)6-s + 1.43·7-s + (0.527 + 0.849i)8-s + 0.760i·9-s + (0.714 + 0.489i)10-s + (−0.919 − 0.919i)11-s + (−0.197 + 0.448i)12-s + (1.25 + 1.25i)13-s + (−0.264 − 1.40i)14-s + 0.424i·15-s + (0.738 − 0.674i)16-s − 0.309·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.423i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.905 + 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.905 + 0.423i$
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: \((2,\ 16,\ (\ :6),\ 0.905 + 0.423i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(1.68275 - 0.374039i\)
\(L(\frac12)\) \(\approx\) \(1.68275 - 0.374039i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (11.7 + 62.9i)T \)
good3 \( 1 + (-252. + 252. i)T - 5.31e5iT^{2} \)
5 \( 1 + (9.56e3 - 9.56e3i)T - 2.44e8iT^{2} \)
7 \( 1 - 1.68e5T + 1.38e10T^{2} \)
11 \( 1 + (1.62e6 + 1.62e6i)T + 3.13e12iT^{2} \)
13 \( 1 + (-6.06e6 - 6.06e6i)T + 2.32e13iT^{2} \)
17 \( 1 + 7.47e6T + 5.82e14T^{2} \)
19 \( 1 + (-6.07e7 + 6.07e7i)T - 2.21e15iT^{2} \)
23 \( 1 + 1.47e7T + 2.19e16T^{2} \)
29 \( 1 + (-6.03e8 - 6.03e8i)T + 3.53e17iT^{2} \)
31 \( 1 + 1.95e8iT - 7.87e17T^{2} \)
37 \( 1 + (-2.13e8 + 2.13e8i)T - 6.58e18iT^{2} \)
41 \( 1 - 7.06e9iT - 2.25e19T^{2} \)
43 \( 1 + (-5.07e9 - 5.07e9i)T + 3.99e19iT^{2} \)
47 \( 1 - 6.27e9iT - 1.16e20T^{2} \)
53 \( 1 + (-3.73e9 + 3.73e9i)T - 4.91e20iT^{2} \)
59 \( 1 + (4.12e8 + 4.12e8i)T + 1.77e21iT^{2} \)
61 \( 1 + (2.53e10 + 2.53e10i)T + 2.65e21iT^{2} \)
67 \( 1 + (-2.76e10 + 2.76e10i)T - 8.18e21iT^{2} \)
71 \( 1 + 2.78e10T + 1.64e22T^{2} \)
73 \( 1 - 3.72e9iT - 2.29e22T^{2} \)
79 \( 1 + 3.15e11iT - 5.90e22T^{2} \)
83 \( 1 + (1.61e11 - 1.61e11i)T - 1.06e23iT^{2} \)
89 \( 1 + 5.89e11iT - 2.46e23T^{2} \)
97 \( 1 + 9.78e11T + 6.93e23T^{2} \)
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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

−16.10514882463023042220354531324, −14.20684606926586375279377302731, −13.44632488874782258463713743919, −11.27578047798109407797437407192, −11.08954308646099733098739482397, −8.693447536481200261729008159801, −7.67303370085194651126111296723, −4.77389754361268279996910508667, −2.92187531113027267058683883620, −1.34222768516416051493592382312, 0.894492869940579156049558500970, 4.08231286798448414558226207337, 5.45047436444570675296916232731, 7.77804784532439666571014367187, 8.517848511420772897065670610544, 10.26708620023079796829379745886, 12.28053795486062683468221176075, 13.95222810611983567365535228724, 15.28030469198985493178634619895, 15.83632077453695946819692574439

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