Properties

Label 2-2e4-16.3-c10-0-5
Degree $2$
Conductor $16$
Sign $-0.380 - 0.924i$
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  + (31.9 + 0.0155i)2-s + (−39.7 + 39.7i)3-s + (1.02e3 + 0.998i)4-s + (−4.09e3 + 4.09e3i)5-s + (−1.27e3 + 1.27e3i)6-s − 7.86e3·7-s + (3.27e4 + 47.9i)8-s + 5.58e4i·9-s + (−1.31e5 + 1.31e5i)10-s + (−4.76e4 − 4.76e4i)11-s + (−4.07e4 + 4.06e4i)12-s + (2.25e5 + 2.25e5i)13-s + (−2.51e5 − 122. i)14-s − 3.25e5i·15-s + (1.04e6 + 2.04e3i)16-s − 7.84e5·17-s + ⋯
L(s)  = 1  + (0.999 + 0.000487i)2-s + (−0.163 + 0.163i)3-s + (0.999 + 0.000974i)4-s + (−1.31 + 1.31i)5-s + (−0.163 + 0.163i)6-s − 0.468·7-s + (0.999 + 0.00146i)8-s + 0.946i·9-s + (−1.31 + 1.31i)10-s + (−0.295 − 0.295i)11-s + (−0.163 + 0.163i)12-s + (0.606 + 0.606i)13-s + (−0.468 − 0.000228i)14-s − 0.428i·15-s + (0.999 + 0.00194i)16-s − 0.552·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.380 - 0.924i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.380 - 0.924i$
Analytic conductor: \(10.1657\)
Root analytic conductor: \(3.18837\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :5),\ -0.380 - 0.924i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.13177 + 1.69024i\)
\(L(\frac12)\) \(\approx\) \(1.13177 + 1.69024i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-31.9 - 0.0155i)T \)
good3 \( 1 + (39.7 - 39.7i)T - 5.90e4iT^{2} \)
5 \( 1 + (4.09e3 - 4.09e3i)T - 9.76e6iT^{2} \)
7 \( 1 + 7.86e3T + 2.82e8T^{2} \)
11 \( 1 + (4.76e4 + 4.76e4i)T + 2.59e10iT^{2} \)
13 \( 1 + (-2.25e5 - 2.25e5i)T + 1.37e11iT^{2} \)
17 \( 1 + 7.84e5T + 2.01e12T^{2} \)
19 \( 1 + (1.71e6 - 1.71e6i)T - 6.13e12iT^{2} \)
23 \( 1 - 1.21e7T + 4.14e13T^{2} \)
29 \( 1 + (-3.70e6 - 3.70e6i)T + 4.20e14iT^{2} \)
31 \( 1 + 3.97e7iT - 8.19e14T^{2} \)
37 \( 1 + (-6.77e6 + 6.77e6i)T - 4.80e15iT^{2} \)
41 \( 1 - 6.98e7iT - 1.34e16T^{2} \)
43 \( 1 + (-1.43e8 - 1.43e8i)T + 2.16e16iT^{2} \)
47 \( 1 - 2.76e8iT - 5.25e16T^{2} \)
53 \( 1 + (1.93e8 - 1.93e8i)T - 1.74e17iT^{2} \)
59 \( 1 + (-4.10e8 - 4.10e8i)T + 5.11e17iT^{2} \)
61 \( 1 + (-2.74e8 - 2.74e8i)T + 7.13e17iT^{2} \)
67 \( 1 + (8.74e8 - 8.74e8i)T - 1.82e18iT^{2} \)
71 \( 1 - 2.68e9T + 3.25e18T^{2} \)
73 \( 1 + 2.43e9iT - 4.29e18T^{2} \)
79 \( 1 + 1.43e8iT - 9.46e18T^{2} \)
83 \( 1 + (2.28e8 - 2.28e8i)T - 1.55e19iT^{2} \)
89 \( 1 - 1.16e9iT - 3.11e19T^{2} \)
97 \( 1 - 4.04e9T + 7.37e19T^{2} \)
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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

−16.49740065054637725159667461485, −15.57163803643705339335745480513, −14.51381513510995836745706549947, −13.07714312226702541882249545209, −11.34695158169468913182527694007, −10.77279787120341828104119947680, −7.74413018006127643865602848178, −6.42592648659972158268685849427, −4.25464838326942057491541221576, −2.83036510563786261583325087610, 0.71680837613946328080426567872, 3.55571871070489205995799203505, 4.98820747967921460178035512581, 6.93594056782735978444787717858, 8.700306275105210190148627088636, 11.15867033965053090058686197624, 12.45971875755720634588484195828, 13.02734906760109279973016749959, 15.23065473167252914910344474982, 15.82768680985611208565388607803

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