Properties

Degree $2$
Conductor $16$
Sign $0.0846 + 0.996i$
Motivic weight $4$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.47 − 3.14i)2-s + (−7.86 − 7.86i)3-s + (−3.78 − 15.5i)4-s + (27.2 + 27.2i)5-s + (−44.1 + 5.30i)6-s + 50.3·7-s + (−58.2 − 26.5i)8-s + 42.8i·9-s + (152. − 18.3i)10-s + (−53.1 + 53.1i)11-s + (−92.5 + 152. i)12-s + (−125. + 125. i)13-s + (124. − 158. i)14-s − 428. i·15-s + (−227. + 117. i)16-s + 286.·17-s + ⋯
L(s)  = 1  + (0.617 − 0.786i)2-s + (−0.874 − 0.874i)3-s + (−0.236 − 0.971i)4-s + (1.08 + 1.08i)5-s + (−1.22 + 0.147i)6-s + 1.02·7-s + (−0.910 − 0.414i)8-s + 0.528i·9-s + (1.52 − 0.183i)10-s + (−0.438 + 0.438i)11-s + (−0.642 + 1.05i)12-s + (−0.740 + 0.740i)13-s + (0.634 − 0.807i)14-s − 1.90i·15-s + (−0.888 + 0.459i)16-s + 0.990·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0846 + 0.996i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0846 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.0846 + 0.996i$
Motivic weight: \(4\)
Character: $\chi_{16} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :2),\ 0.0846 + 0.996i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.02804 - 0.944381i\)
\(L(\frac12)\) \(\approx\) \(1.02804 - 0.944381i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.47 + 3.14i)T \)
good3 \( 1 + (7.86 + 7.86i)T + 81iT^{2} \)
5 \( 1 + (-27.2 - 27.2i)T + 625iT^{2} \)
7 \( 1 - 50.3T + 2.40e3T^{2} \)
11 \( 1 + (53.1 - 53.1i)T - 1.46e4iT^{2} \)
13 \( 1 + (125. - 125. i)T - 2.85e4iT^{2} \)
17 \( 1 - 286.T + 8.35e4T^{2} \)
19 \( 1 + (99.5 + 99.5i)T + 1.30e5iT^{2} \)
23 \( 1 + 100.T + 2.79e5T^{2} \)
29 \( 1 + (-343. + 343. i)T - 7.07e5iT^{2} \)
31 \( 1 - 208. iT - 9.23e5T^{2} \)
37 \( 1 + (1.15e3 + 1.15e3i)T + 1.87e6iT^{2} \)
41 \( 1 + 2.33e3iT - 2.82e6T^{2} \)
43 \( 1 + (2.07e3 - 2.07e3i)T - 3.41e6iT^{2} \)
47 \( 1 + 1.05e3iT - 4.87e6T^{2} \)
53 \( 1 + (-2.13e3 - 2.13e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (-3.72e3 + 3.72e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (-2.49e3 + 2.49e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (329. + 329. i)T + 2.01e7iT^{2} \)
71 \( 1 + 1.04e3T + 2.54e7T^{2} \)
73 \( 1 - 2.67e3iT - 2.83e7T^{2} \)
79 \( 1 - 4.47e3iT - 3.89e7T^{2} \)
83 \( 1 + (-1.45e3 - 1.45e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 1.14e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.31e4T + 8.85e7T^{2} \)
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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

−18.13331388844150196708837995523, −17.41787396203579471655084913685, −14.76522579927562464605458801098, −13.88270684440660163084731792241, −12.39033133761968601624984824071, −11.24862285803189476511894331634, −10.01300273275767311285942412036, −6.87198751507746545371523975885, −5.40183618026703004892783855480, −1.99332759993399153954160636058, 4.92363426132975832252453235095, 5.57777607942541232102608291017, 8.276255223790921748716519693415, 10.11164253760712221901792607020, 11.96364227870492534626312877646, 13.38362949683296981123464742261, 14.80503495352825476353317421586, 16.30213826808910014846143893822, 17.03298529416628319144713373686, 17.85821045957825609090904445994

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