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Properties

Label 2-2e4-16.11-c12-0-0
Degree $2$
Conductor $16$
Sign $-0.638 + 0.769i$
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  + (41.7 + 48.5i)2-s + (193. + 193. i)3-s + (−612. + 4.04e3i)4-s + (−1.03e4 − 1.03e4i)5-s + (−1.31e3 + 1.74e4i)6-s − 1.08e5·7-s + (−2.22e5 + 1.39e5i)8-s − 4.56e5i·9-s + (7.02e4 − 9.34e5i)10-s + (−1.04e6 + 1.04e6i)11-s + (−9.01e5 + 6.64e5i)12-s + (−1.27e6 + 1.27e6i)13-s + (−4.52e6 − 5.26e6i)14-s − 4.00e6i·15-s + (−1.60e7 − 4.96e6i)16-s + 1.95e7·17-s + ⋯
L(s)  = 1  + (0.652 + 0.758i)2-s + (0.265 + 0.265i)3-s + (−0.149 + 0.988i)4-s + (−0.662 − 0.662i)5-s + (−0.0281 + 0.373i)6-s − 0.921·7-s + (−0.847 + 0.531i)8-s − 0.859i·9-s + (0.0702 − 0.934i)10-s + (−0.590 + 0.590i)11-s + (−0.301 + 0.222i)12-s + (−0.264 + 0.264i)13-s + (−0.601 − 0.698i)14-s − 0.351i·15-s + (−0.955 − 0.295i)16-s + 0.808·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.638 + 0.769i$
Analytic conductor: \(14.6239\)
Root analytic conductor: \(3.82412\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :6),\ -0.638 + 0.769i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.106038 - 0.225876i\)
\(L(\frac12)\) \(\approx\) \(0.106038 - 0.225876i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-41.7 - 48.5i)T \)
good3 \( 1 + (-193. - 193. i)T + 5.31e5iT^{2} \)
5 \( 1 + (1.03e4 + 1.03e4i)T + 2.44e8iT^{2} \)
7 \( 1 + 1.08e5T + 1.38e10T^{2} \)
11 \( 1 + (1.04e6 - 1.04e6i)T - 3.13e12iT^{2} \)
13 \( 1 + (1.27e6 - 1.27e6i)T - 2.32e13iT^{2} \)
17 \( 1 - 1.95e7T + 5.82e14T^{2} \)
19 \( 1 + (2.72e7 + 2.72e7i)T + 2.21e15iT^{2} \)
23 \( 1 + 1.76e8T + 2.19e16T^{2} \)
29 \( 1 + (3.27e8 - 3.27e8i)T - 3.53e17iT^{2} \)
31 \( 1 - 1.40e9iT - 7.87e17T^{2} \)
37 \( 1 + (-3.08e9 - 3.08e9i)T + 6.58e18iT^{2} \)
41 \( 1 - 1.10e9iT - 2.25e19T^{2} \)
43 \( 1 + (4.13e9 - 4.13e9i)T - 3.99e19iT^{2} \)
47 \( 1 - 7.39e9iT - 1.16e20T^{2} \)
53 \( 1 + (2.73e10 + 2.73e10i)T + 4.91e20iT^{2} \)
59 \( 1 + (-4.52e9 + 4.52e9i)T - 1.77e21iT^{2} \)
61 \( 1 + (-2.45e9 + 2.45e9i)T - 2.65e21iT^{2} \)
67 \( 1 + (2.12e9 + 2.12e9i)T + 8.18e21iT^{2} \)
71 \( 1 - 5.19e10T + 1.64e22T^{2} \)
73 \( 1 + 2.15e11iT - 2.29e22T^{2} \)
79 \( 1 + 3.83e10iT - 5.90e22T^{2} \)
83 \( 1 + (-3.95e11 - 3.95e11i)T + 1.06e23iT^{2} \)
89 \( 1 + 7.22e11iT - 2.46e23T^{2} \)
97 \( 1 + 1.01e12T + 6.93e23T^{2} \)
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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

−16.48114016344170639063822865050, −15.69680890752411944948996875461, −14.57116121723174566773114379652, −12.89473611880678801085583688351, −12.08964716791311844286813044268, −9.585817192630555973755078012716, −8.131184836235198192487687792549, −6.52691604536041484790682008318, −4.65040796180522670387634452968, −3.26542536854695070391282484714, 0.07441663744463600016696007147, 2.42459839448482057215573996888, 3.72761870438839257011962777326, 5.85598680039794639302993500228, 7.76247969518470639325254233723, 9.964427772105536416637700926032, 11.14700583490512884244608134886, 12.62919952759020062749342348915, 13.72245379660269972690997169280, 15.01957787423387153830618311346

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