Properties

Label 2-2e4-4.3-c10-0-4
Degree $2$
Conductor $16$
Sign $-0.500 + 0.866i$
Analytic cond. $10.1657$
Root an. cond. $3.18837$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 408. i·3-s + 5.36e3·5-s − 9.54e3i·7-s − 1.07e5·9-s − 1.01e5i·11-s + 3.74e4·13-s − 2.19e6i·15-s − 1.39e6·17-s + 2.17e6i·19-s − 3.89e6·21-s − 4.54e6i·23-s + 1.90e7·25-s + 1.98e7i·27-s + 7.75e6·29-s + 4.37e7i·31-s + ⋯
L(s)  = 1  − 1.68i·3-s + 1.71·5-s − 0.567i·7-s − 1.82·9-s − 0.629i·11-s + 0.100·13-s − 2.88i·15-s − 0.985·17-s + 0.877i·19-s − 0.954·21-s − 0.705i·23-s + 1.94·25-s + 1.38i·27-s + 0.378·29-s + 1.52i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.500 + 0.866i$
Analytic conductor: \(10.1657\)
Root analytic conductor: \(3.18837\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :5),\ -0.500 + 0.866i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.04620 - 1.81207i\)
\(L(\frac12)\) \(\approx\) \(1.04620 - 1.81207i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 408. iT - 5.90e4T^{2} \)
5 \( 1 - 5.36e3T + 9.76e6T^{2} \)
7 \( 1 + 9.54e3iT - 2.82e8T^{2} \)
11 \( 1 + 1.01e5iT - 2.59e10T^{2} \)
13 \( 1 - 3.74e4T + 1.37e11T^{2} \)
17 \( 1 + 1.39e6T + 2.01e12T^{2} \)
19 \( 1 - 2.17e6iT - 6.13e12T^{2} \)
23 \( 1 + 4.54e6iT - 4.14e13T^{2} \)
29 \( 1 - 7.75e6T + 4.20e14T^{2} \)
31 \( 1 - 4.37e7iT - 8.19e14T^{2} \)
37 \( 1 - 2.45e7T + 4.80e15T^{2} \)
41 \( 1 - 1.44e8T + 1.34e16T^{2} \)
43 \( 1 + 2.85e8iT - 2.16e16T^{2} \)
47 \( 1 - 5.16e6iT - 5.25e16T^{2} \)
53 \( 1 - 2.90e8T + 1.74e17T^{2} \)
59 \( 1 - 7.53e8iT - 5.11e17T^{2} \)
61 \( 1 + 5.34e8T + 7.13e17T^{2} \)
67 \( 1 - 8.34e8iT - 1.82e18T^{2} \)
71 \( 1 + 1.40e9iT - 3.25e18T^{2} \)
73 \( 1 - 1.64e9T + 4.29e18T^{2} \)
79 \( 1 - 2.71e9iT - 9.46e18T^{2} \)
83 \( 1 - 4.09e9iT - 1.55e19T^{2} \)
89 \( 1 + 4.62e9T + 3.11e19T^{2} \)
97 \( 1 - 2.28e9T + 7.37e19T^{2} \)
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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

−16.78812280568250263333322473177, −14.10681470129994491281575411642, −13.58330032109512062695845732692, −12.48131386074017299369512442675, −10.58286418060353812232858296586, −8.700346315919732513757505959186, −6.89964890220859404897507991748, −5.84958137220950197859296000347, −2.31777551149037361156197018100, −1.04463879815843202073031552507, 2.45967805455503693161608709367, 4.68758227200634109429992512006, 5.98059704988160615002328850388, 9.147324399292433709100372209070, 9.755150154559795966936835243180, 11.07874599739570773075978228905, 13.27691570971726492594134170083, 14.67418699120793804235137948633, 15.70680270759145709406398360734, 17.08754408970829694858334836338

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