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Properties

Label 2-2e4-16.11-c12-0-3
Degree $2$
Conductor $16$
Sign $0.958 - 0.286i$
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  + (14.0 − 62.4i)2-s + (−832. − 832. i)3-s + (−3.69e3 − 1.75e3i)4-s + (1.76e4 + 1.76e4i)5-s + (−6.36e4 + 4.02e4i)6-s − 1.64e5·7-s + (−1.61e5 + 2.06e5i)8-s + 8.53e5i·9-s + (1.34e6 − 8.51e5i)10-s + (1.61e6 − 1.61e6i)11-s + (1.61e6 + 4.54e6i)12-s + (−1.20e6 + 1.20e6i)13-s + (−2.31e6 + 1.02e7i)14-s − 2.93e7i·15-s + (1.05e7 + 1.30e7i)16-s − 6.99e6·17-s + ⋯
L(s)  = 1  + (0.220 − 0.975i)2-s + (−1.14 − 1.14i)3-s + (−0.903 − 0.429i)4-s + (1.12 + 1.12i)5-s + (−1.36 + 0.862i)6-s − 1.39·7-s + (−0.617 + 0.786i)8-s + 1.60i·9-s + (1.34 − 0.851i)10-s + (0.910 − 0.910i)11-s + (0.540 + 1.52i)12-s + (−0.250 + 0.250i)13-s + (−0.306 + 1.36i)14-s − 2.57i·15-s + (0.631 + 0.775i)16-s − 0.289·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.286i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.958 - 0.286i$
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: \((2,\ 16,\ (\ :6),\ 0.958 - 0.286i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.615454 + 0.0900151i\)
\(L(\frac12)\) \(\approx\) \(0.615454 + 0.0900151i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-14.0 + 62.4i)T \)
good3 \( 1 + (832. + 832. i)T + 5.31e5iT^{2} \)
5 \( 1 + (-1.76e4 - 1.76e4i)T + 2.44e8iT^{2} \)
7 \( 1 + 1.64e5T + 1.38e10T^{2} \)
11 \( 1 + (-1.61e6 + 1.61e6i)T - 3.13e12iT^{2} \)
13 \( 1 + (1.20e6 - 1.20e6i)T - 2.32e13iT^{2} \)
17 \( 1 + 6.99e6T + 5.82e14T^{2} \)
19 \( 1 + (-2.17e7 - 2.17e7i)T + 2.21e15iT^{2} \)
23 \( 1 + 6.39e7T + 2.19e16T^{2} \)
29 \( 1 + (7.89e7 - 7.89e7i)T - 3.53e17iT^{2} \)
31 \( 1 - 1.11e9iT - 7.87e17T^{2} \)
37 \( 1 + (-2.71e9 - 2.71e9i)T + 6.58e18iT^{2} \)
41 \( 1 + 4.18e9iT - 2.25e19T^{2} \)
43 \( 1 + (5.33e9 - 5.33e9i)T - 3.99e19iT^{2} \)
47 \( 1 - 1.47e10iT - 1.16e20T^{2} \)
53 \( 1 + (4.02e9 + 4.02e9i)T + 4.91e20iT^{2} \)
59 \( 1 + (3.28e10 - 3.28e10i)T - 1.77e21iT^{2} \)
61 \( 1 + (2.15e10 - 2.15e10i)T - 2.65e21iT^{2} \)
67 \( 1 + (8.97e10 + 8.97e10i)T + 8.18e21iT^{2} \)
71 \( 1 - 1.25e11T + 1.64e22T^{2} \)
73 \( 1 - 1.89e11iT - 2.29e22T^{2} \)
79 \( 1 + 1.73e11iT - 5.90e22T^{2} \)
83 \( 1 + (7.00e10 + 7.00e10i)T + 1.06e23iT^{2} \)
89 \( 1 + 9.79e10iT - 2.46e23T^{2} \)
97 \( 1 + 1.25e12T + 6.93e23T^{2} \)
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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

−16.74930846649611889086411921009, −14.14168979538237104132262464937, −13.27064876402947331632968135453, −12.06643146893531647118337869596, −10.84374671319811650111598125495, −9.604459055302980503688810407607, −6.58641319829139012008463458035, −5.92173133447877426003771315357, −3.00366914648043694767679699555, −1.37257562489699707278727775116, 0.29077824166302262000390358881, 4.21267444521425949990494431344, 5.44112961253040435642864996396, 6.45512208158489790118467243417, 9.414542600276139200888675152698, 9.732174966661103885970671180762, 12.25421279468277019171891158143, 13.34669968360881815801355498325, 15.22645405230484328924193636538, 16.40389217872068476850435860307

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