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Properties

Label 2-2e4-16.13-c13-0-4
Degree $2$
Conductor $16$
Sign $0.627 + 0.778i$
Analytic cond. $17.1569$
Root an. cond. $4.14209$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−40.5 − 80.9i)2-s + (−1.63e3 − 1.63e3i)3-s + (−4.90e3 + 6.55e3i)4-s + (9.55e3 − 9.55e3i)5-s + (−6.62e4 + 1.99e5i)6-s + 1.33e5i·7-s + (7.29e5 + 1.31e5i)8-s + 3.78e6i·9-s + (−1.16e6 − 3.86e5i)10-s + (−3.83e5 + 3.83e5i)11-s + (1.88e7 − 2.70e6i)12-s + (−1.38e6 − 1.38e6i)13-s + (1.08e7 − 5.42e6i)14-s − 3.13e7·15-s + (−1.89e7 − 6.43e7i)16-s − 1.51e8·17-s + ⋯
L(s)  = 1  + (−0.447 − 0.894i)2-s + (−1.29 − 1.29i)3-s + (−0.599 + 0.800i)4-s + (0.273 − 0.273i)5-s + (−0.579 + 1.74i)6-s + 0.430i·7-s + (0.984 + 0.177i)8-s + 2.37i·9-s + (−0.367 − 0.122i)10-s + (−0.0651 + 0.0651i)11-s + (1.81 − 0.261i)12-s + (−0.0794 − 0.0794i)13-s + (0.384 − 0.192i)14-s − 0.710·15-s + (−0.282 − 0.959i)16-s − 1.52·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.627 + 0.778i$
Analytic conductor: \(17.1569\)
Root analytic conductor: \(4.14209\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :13/2),\ 0.627 + 0.778i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.579969 - 0.277324i\)
\(L(\frac12)\) \(\approx\) \(0.579969 - 0.277324i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (40.5 + 80.9i)T \)
good3 \( 1 + (1.63e3 + 1.63e3i)T + 1.59e6iT^{2} \)
5 \( 1 + (-9.55e3 + 9.55e3i)T - 1.22e9iT^{2} \)
7 \( 1 - 1.33e5iT - 9.68e10T^{2} \)
11 \( 1 + (3.83e5 - 3.83e5i)T - 3.45e13iT^{2} \)
13 \( 1 + (1.38e6 + 1.38e6i)T + 3.02e14iT^{2} \)
17 \( 1 + 1.51e8T + 9.90e15T^{2} \)
19 \( 1 + (-1.91e8 - 1.91e8i)T + 4.20e16iT^{2} \)
23 \( 1 + 1.33e9iT - 5.04e17T^{2} \)
29 \( 1 + (-2.13e9 - 2.13e9i)T + 1.02e19iT^{2} \)
31 \( 1 - 3.46e9T + 2.44e19T^{2} \)
37 \( 1 + (2.02e10 - 2.02e10i)T - 2.43e20iT^{2} \)
41 \( 1 - 5.32e9iT - 9.25e20T^{2} \)
43 \( 1 + (-2.42e10 + 2.42e10i)T - 1.71e21iT^{2} \)
47 \( 1 + 4.54e10T + 5.46e21T^{2} \)
53 \( 1 + (-1.29e11 + 1.29e11i)T - 2.60e22iT^{2} \)
59 \( 1 + (-2.31e11 + 2.31e11i)T - 1.04e23iT^{2} \)
61 \( 1 + (-3.36e11 - 3.36e11i)T + 1.61e23iT^{2} \)
67 \( 1 + (-1.18e11 - 1.18e11i)T + 5.48e23iT^{2} \)
71 \( 1 - 8.45e10iT - 1.16e24T^{2} \)
73 \( 1 + 1.06e12iT - 1.67e24T^{2} \)
79 \( 1 - 6.11e11T + 4.66e24T^{2} \)
83 \( 1 + (-1.46e12 - 1.46e12i)T + 8.87e24iT^{2} \)
89 \( 1 - 6.17e12iT - 2.19e25T^{2} \)
97 \( 1 + 4.68e12T + 6.73e25T^{2} \)
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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

−16.39358587891200299972900832027, −13.60395311066141452675026877358, −12.56696790638628444595137918547, −11.73156879676566519837322315698, −10.46290399693472273332166097801, −8.455838776304165910382003446999, −6.78684551647745793048191370456, −5.06378783553280544960006403550, −2.19001466329150210901052731783, −0.900311695348686638962458485171, 0.47861118784644370996406140413, 4.30090350667297323087316800601, 5.54523036861692323417358904745, 6.86274276996307423353203501685, 9.229786930626711775196181006346, 10.30143243866581864750569665432, 11.39106416303115251897199967758, 13.76908834033856347383554370814, 15.46136837888685929936094106903, 15.99109785708743308532065897910

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