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Properties

Label 2-2e4-16.13-c21-0-37
Degree $2$
Conductor $16$
Sign $-0.758 - 0.651i$
Analytic cond. $44.7163$
Root an. cond. $6.68703$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (394. − 1.39e3i)2-s + (7.32e4 + 7.32e4i)3-s + (−1.78e6 − 1.09e6i)4-s + (2.01e7 − 2.01e7i)5-s + (1.30e8 − 7.31e7i)6-s + 9.05e8i·7-s + (−2.23e9 + 2.05e9i)8-s + 2.66e8i·9-s + (−2.01e10 − 3.59e10i)10-s + (−1.00e11 + 1.00e11i)11-s + (−5.04e10 − 2.11e11i)12-s + (−5.79e11 − 5.79e11i)13-s + (1.26e12 + 3.56e11i)14-s + 2.94e12·15-s + (1.98e12 + 3.92e12i)16-s − 6.28e12·17-s + ⋯
L(s)  = 1  + (0.272 − 0.962i)2-s + (0.716 + 0.716i)3-s + (−0.851 − 0.523i)4-s + (0.921 − 0.921i)5-s + (0.883 − 0.494i)6-s + 1.21i·7-s + (−0.735 + 0.677i)8-s + 0.0255i·9-s + (−0.636 − 1.13i)10-s + (−1.17 + 1.17i)11-s + (−0.235 − 0.985i)12-s + (−1.16 − 1.16i)13-s + (1.16 + 0.329i)14-s + 1.31·15-s + (0.451 + 0.892i)16-s − 0.756·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.758 - 0.651i$
Analytic conductor: \(44.7163\)
Root analytic conductor: \(6.68703\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :21/2),\ -0.758 - 0.651i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.2191488437\)
\(L(\frac12)\) \(\approx\) \(0.2191488437\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-394. + 1.39e3i)T \)
good3 \( 1 + (-7.32e4 - 7.32e4i)T + 1.04e10iT^{2} \)
5 \( 1 + (-2.01e7 + 2.01e7i)T - 4.76e14iT^{2} \)
7 \( 1 - 9.05e8iT - 5.58e17T^{2} \)
11 \( 1 + (1.00e11 - 1.00e11i)T - 7.40e21iT^{2} \)
13 \( 1 + (5.79e11 + 5.79e11i)T + 2.47e23iT^{2} \)
17 \( 1 + 6.28e12T + 6.90e25T^{2} \)
19 \( 1 + (-6.00e12 - 6.00e12i)T + 7.14e26iT^{2} \)
23 \( 1 + 3.34e14iT - 3.94e28T^{2} \)
29 \( 1 + (1.37e15 + 1.37e15i)T + 5.13e30iT^{2} \)
31 \( 1 + 7.62e15T + 2.08e31T^{2} \)
37 \( 1 + (3.59e16 - 3.59e16i)T - 8.55e32iT^{2} \)
41 \( 1 - 4.89e16iT - 7.38e33T^{2} \)
43 \( 1 + (-1.77e16 + 1.77e16i)T - 2.00e34iT^{2} \)
47 \( 1 + 4.72e16T + 1.30e35T^{2} \)
53 \( 1 + (7.77e17 - 7.77e17i)T - 1.62e36iT^{2} \)
59 \( 1 + (-1.94e18 + 1.94e18i)T - 1.54e37iT^{2} \)
61 \( 1 + (1.24e17 + 1.24e17i)T + 3.10e37iT^{2} \)
67 \( 1 + (-3.64e18 - 3.64e18i)T + 2.22e38iT^{2} \)
71 \( 1 - 4.13e19iT - 7.52e38T^{2} \)
73 \( 1 + 8.97e18iT - 1.34e39T^{2} \)
79 \( 1 + 5.58e19T + 7.08e39T^{2} \)
83 \( 1 + (-2.29e19 - 2.29e19i)T + 1.99e40iT^{2} \)
89 \( 1 + 1.06e20iT - 8.65e40T^{2} \)
97 \( 1 - 4.56e20T + 5.27e41T^{2} \)
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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

−12.95190571591294347978920197409, −12.40200785584474090299747738688, −10.22958323449490641578026882358, −9.489389231135300501638199856545, −8.488517346787285380080576405963, −5.46194073133274482831711119344, −4.69531766420158683601768280971, −2.76209899195238445396506677988, −2.04766550292692616566463222745, −0.04355664833088606091054804237, 2.04730999600144339565080303032, 3.42949532836299735594349288751, 5.35954256618222035029652342010, 6.98715758009857349044147525933, 7.51985954636334550691434318881, 9.211045304107942607330343394882, 10.77013330650678515816388120596, 13.18515492272095730327226411706, 13.80486764717644369185518171119, 14.44773180734994661038967984964

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