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Properties

Label 2-2e4-16.11-c2-0-0
Degree $2$
Conductor $16$
Sign $0.718 - 0.695i$
Analytic cond. $0.435968$
Root an. cond. $0.660279$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.55 + 1.26i)2-s + (2.10 + 2.10i)3-s + (0.813 − 3.91i)4-s + (−4.62 − 4.62i)5-s + (−5.91 − 0.608i)6-s + 3.04·7-s + (3.68 + 7.10i)8-s − 0.156i·9-s + (13.0 + 1.33i)10-s + (−9.15 + 9.15i)11-s + (9.94 − 6.52i)12-s + (−5.78 + 5.78i)13-s + (−4.72 + 3.84i)14-s − 19.4i·15-s + (−14.6 − 6.37i)16-s + 17.6·17-s + ⋯
L(s)  = 1  + (−0.775 + 0.631i)2-s + (0.700 + 0.700i)3-s + (0.203 − 0.979i)4-s + (−0.925 − 0.925i)5-s + (−0.986 − 0.101i)6-s + 0.435·7-s + (0.460 + 0.887i)8-s − 0.0174i·9-s + (1.30 + 0.133i)10-s + (−0.831 + 0.831i)11-s + (0.828 − 0.543i)12-s + (−0.444 + 0.444i)13-s + (−0.337 + 0.274i)14-s − 1.29i·15-s + (−0.917 − 0.398i)16-s + 1.03·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.718 - 0.695i$
Analytic conductor: \(0.435968\)
Root analytic conductor: \(0.660279\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :1),\ 0.718 - 0.695i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.595321 + 0.240695i\)
\(L(\frac12)\) \(\approx\) \(0.595321 + 0.240695i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.55 - 1.26i)T \)
good3 \( 1 + (-2.10 - 2.10i)T + 9iT^{2} \)
5 \( 1 + (4.62 + 4.62i)T + 25iT^{2} \)
7 \( 1 - 3.04T + 49T^{2} \)
11 \( 1 + (9.15 - 9.15i)T - 121iT^{2} \)
13 \( 1 + (5.78 - 5.78i)T - 169iT^{2} \)
17 \( 1 - 17.6T + 289T^{2} \)
19 \( 1 + (1.15 + 1.15i)T + 361iT^{2} \)
23 \( 1 + 3.45T + 529T^{2} \)
29 \( 1 + (-12.1 + 12.1i)T - 841iT^{2} \)
31 \( 1 - 38.5iT - 961T^{2} \)
37 \( 1 + (0.0972 + 0.0972i)T + 1.36e3iT^{2} \)
41 \( 1 + 51.5iT - 1.68e3T^{2} \)
43 \( 1 + (1.70 - 1.70i)T - 1.84e3iT^{2} \)
47 \( 1 - 24.1iT - 2.20e3T^{2} \)
53 \( 1 + (-27.0 - 27.0i)T + 2.80e3iT^{2} \)
59 \( 1 + (-19.5 + 19.5i)T - 3.48e3iT^{2} \)
61 \( 1 + (-16.7 + 16.7i)T - 3.72e3iT^{2} \)
67 \( 1 + (75.8 + 75.8i)T + 4.48e3iT^{2} \)
71 \( 1 + 134.T + 5.04e3T^{2} \)
73 \( 1 - 112. iT - 5.32e3T^{2} \)
79 \( 1 + 135. iT - 6.24e3T^{2} \)
83 \( 1 + (-74.9 - 74.9i)T + 6.88e3iT^{2} \)
89 \( 1 + 31.4iT - 7.92e3T^{2} \)
97 \( 1 - 31.5T + 9.40e3T^{2} \)
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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

−19.24631576833940179260032195605, −17.69273754984693331912412168442, −16.27321399969846010193682678422, −15.44155438872578373344285833964, −14.38367411776500791876881749275, −12.12711696639815969211762356962, −10.14434279059121646655929681001, −8.811955731886072562243428547535, −7.66736938703655029869243953012, −4.75617402699940195329251308925, 2.98931204697392862733797842527, 7.49795545202933616860365622989, 8.200020929716778120328955933920, 10.43946517157300583994667948221, 11.66918127289927349514823795935, 13.21334748146507348543615789638, 14.76304392848751706692529374921, 16.35332775477675652668389363409, 18.15104896815027068919475755278, 18.93018841176314664890974775170

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