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Properties

Label 2-2e4-16.13-c13-0-18
Degree $2$
Conductor $16$
Sign $0.695 + 0.718i$
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  + (79.9 − 42.3i)2-s + (805. + 805. i)3-s + (4.60e3 − 6.77e3i)4-s + (1.88e4 − 1.88e4i)5-s + (9.85e4 + 3.03e4i)6-s + 1.73e4i·7-s + (8.18e4 − 7.36e5i)8-s − 2.95e5i·9-s + (7.08e5 − 2.30e6i)10-s + (8.32e5 − 8.32e5i)11-s + (9.17e6 − 1.74e6i)12-s + (1.44e7 + 1.44e7i)13-s + (7.33e5 + 1.38e6i)14-s + 3.03e7·15-s + (−2.46e7 − 6.24e7i)16-s + 6.64e7·17-s + ⋯
L(s)  = 1  + (0.883 − 0.467i)2-s + (0.638 + 0.638i)3-s + (0.562 − 0.826i)4-s + (0.538 − 0.538i)5-s + (0.862 + 0.265i)6-s + 0.0556i·7-s + (0.110 − 0.993i)8-s − 0.185i·9-s + (0.224 − 0.728i)10-s + (0.141 − 0.141i)11-s + (0.886 − 0.168i)12-s + (0.831 + 0.831i)13-s + (0.0260 + 0.0492i)14-s + 0.687·15-s + (−0.367 − 0.930i)16-s + 0.667·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.695 + 0.718i$
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: \((2,\ 16,\ (\ :13/2),\ 0.695 + 0.718i)\)

Particular Values

\(L(7)\) \(\approx\) \(3.95337 - 1.67621i\)
\(L(\frac12)\) \(\approx\) \(3.95337 - 1.67621i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-79.9 + 42.3i)T \)
good3 \( 1 + (-805. - 805. i)T + 1.59e6iT^{2} \)
5 \( 1 + (-1.88e4 + 1.88e4i)T - 1.22e9iT^{2} \)
7 \( 1 - 1.73e4iT - 9.68e10T^{2} \)
11 \( 1 + (-8.32e5 + 8.32e5i)T - 3.45e13iT^{2} \)
13 \( 1 + (-1.44e7 - 1.44e7i)T + 3.02e14iT^{2} \)
17 \( 1 - 6.64e7T + 9.90e15T^{2} \)
19 \( 1 + (6.64e7 + 6.64e7i)T + 4.20e16iT^{2} \)
23 \( 1 + 3.89e8iT - 5.04e17T^{2} \)
29 \( 1 + (6.97e8 + 6.97e8i)T + 1.02e19iT^{2} \)
31 \( 1 - 1.62e9T + 2.44e19T^{2} \)
37 \( 1 + (2.11e10 - 2.11e10i)T - 2.43e20iT^{2} \)
41 \( 1 - 4.22e10iT - 9.25e20T^{2} \)
43 \( 1 + (3.37e10 - 3.37e10i)T - 1.71e21iT^{2} \)
47 \( 1 + 6.25e10T + 5.46e21T^{2} \)
53 \( 1 + (1.16e10 - 1.16e10i)T - 2.60e22iT^{2} \)
59 \( 1 + (3.74e11 - 3.74e11i)T - 1.04e23iT^{2} \)
61 \( 1 + (1.50e10 + 1.50e10i)T + 1.61e23iT^{2} \)
67 \( 1 + (-3.31e11 - 3.31e11i)T + 5.48e23iT^{2} \)
71 \( 1 + 1.41e12iT - 1.16e24T^{2} \)
73 \( 1 - 3.43e11iT - 1.67e24T^{2} \)
79 \( 1 + 2.30e12T + 4.66e24T^{2} \)
83 \( 1 + (2.56e12 + 2.56e12i)T + 8.87e24iT^{2} \)
89 \( 1 + 3.79e12iT - 2.19e25T^{2} \)
97 \( 1 - 9.19e12T + 6.73e25T^{2} \)
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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

−15.40032432037402402644655184095, −14.26863883046432546663167965463, −13.20402002538056009651132766470, −11.71338441024596960613702788924, −10.06907388177970434055210984581, −8.888458169815744112981996343493, −6.29236407171435000070656500844, −4.61215871110015316353600766747, −3.22255912866698159047281227846, −1.40938890160941231365386514279, 1.96917510172825294100985517470, 3.40600804941579608052018171080, 5.61104579244252221639803801597, 7.08332563972949674588387811691, 8.349846217335965634807825599122, 10.64629685084155257909623854485, 12.47207866986326649162547089260, 13.63013897276770948979421867489, 14.38010174882765779656998360388, 15.75921998275866267789673825460

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