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Properties

Label 2-2e4-16.13-c13-0-17
Degree $2$
Conductor $16$
Sign $0.00489 + 0.999i$
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  + (−86.5 + 26.3i)2-s + (913. + 913. i)3-s + (6.80e3 − 4.56e3i)4-s + (1.57e4 − 1.57e4i)5-s + (−1.03e5 − 5.50e4i)6-s − 3.56e5i·7-s + (−4.68e5 + 5.74e5i)8-s + 7.60e4i·9-s + (−9.46e5 + 1.77e6i)10-s + (−5.47e6 + 5.47e6i)11-s + (1.03e7 + 2.04e6i)12-s + (−1.13e7 − 1.13e7i)13-s + (9.39e6 + 3.08e7i)14-s + 2.87e7·15-s + (2.53e7 − 6.21e7i)16-s − 4.82e7·17-s + ⋯
L(s)  = 1  + (−0.956 + 0.291i)2-s + (0.723 + 0.723i)3-s + (0.830 − 0.557i)4-s + (0.449 − 0.449i)5-s + (−0.903 − 0.481i)6-s − 1.14i·7-s + (−0.631 + 0.775i)8-s + 0.0476i·9-s + (−0.299 + 0.561i)10-s + (−0.931 + 0.931i)11-s + (1.00 + 0.197i)12-s + (−0.649 − 0.649i)13-s + (0.333 + 1.09i)14-s + 0.651·15-s + (0.378 − 0.925i)16-s − 0.484·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00489 + 0.999i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.00489 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.00489 + 0.999i$
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: \((2,\ 16,\ (\ :13/2),\ 0.00489 + 0.999i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.665417 - 0.662169i\)
\(L(\frac12)\) \(\approx\) \(0.665417 - 0.662169i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (86.5 - 26.3i)T \)
good3 \( 1 + (-913. - 913. i)T + 1.59e6iT^{2} \)
5 \( 1 + (-1.57e4 + 1.57e4i)T - 1.22e9iT^{2} \)
7 \( 1 + 3.56e5iT - 9.68e10T^{2} \)
11 \( 1 + (5.47e6 - 5.47e6i)T - 3.45e13iT^{2} \)
13 \( 1 + (1.13e7 + 1.13e7i)T + 3.02e14iT^{2} \)
17 \( 1 + 4.82e7T + 9.90e15T^{2} \)
19 \( 1 + (2.66e8 + 2.66e8i)T + 4.20e16iT^{2} \)
23 \( 1 + 7.76e8iT - 5.04e17T^{2} \)
29 \( 1 + (-6.91e8 - 6.91e8i)T + 1.02e19iT^{2} \)
31 \( 1 - 1.92e9T + 2.44e19T^{2} \)
37 \( 1 + (-4.08e9 + 4.08e9i)T - 2.43e20iT^{2} \)
41 \( 1 + 7.51e9iT - 9.25e20T^{2} \)
43 \( 1 + (-5.30e8 + 5.30e8i)T - 1.71e21iT^{2} \)
47 \( 1 + 6.49e10T + 5.46e21T^{2} \)
53 \( 1 + (-1.64e11 + 1.64e11i)T - 2.60e22iT^{2} \)
59 \( 1 + (-1.37e11 + 1.37e11i)T - 1.04e23iT^{2} \)
61 \( 1 + (8.73e10 + 8.73e10i)T + 1.61e23iT^{2} \)
67 \( 1 + (9.56e11 + 9.56e11i)T + 5.48e23iT^{2} \)
71 \( 1 - 1.97e12iT - 1.16e24T^{2} \)
73 \( 1 - 2.14e12iT - 1.67e24T^{2} \)
79 \( 1 + 6.56e11T + 4.66e24T^{2} \)
83 \( 1 + (-2.09e12 - 2.09e12i)T + 8.87e24iT^{2} \)
89 \( 1 + 6.04e12iT - 2.19e25T^{2} \)
97 \( 1 - 5.74e12T + 6.73e25T^{2} \)
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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

−15.59942757376916055424964922195, −14.68946894937730779496484939743, −12.97827213265828735657151753703, −10.62210565325073686094091495042, −9.791555660326998501361665650230, −8.503602168901020984825594254374, −6.99356903850022080486895785120, −4.70765015338230226516595351289, −2.45297505233538578131822577228, −0.41010097262320604172598121296, 1.92937833334679246962414498001, 2.74346616177254110434750105486, 6.17944368175163233914743206836, 7.85202571088545176546412608145, 8.865359981497412741752512452146, 10.44388248955320651806464107028, 12.03031154418325097971582838977, 13.40633246541834871078626382309, 14.94678198807873016735157592387, 16.41760377110023135647680947487

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