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Properties

Label 2-2e4-16.13-c9-0-14
Degree $2$
Conductor $16$
Sign $-0.999 - 0.0210i$
Analytic cond. $8.24057$
Root an. cond. $2.87063$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−22.5 − 2.33i)2-s + (−82.6 − 82.6i)3-s + (501. + 105. i)4-s + (1.21e3 − 1.21e3i)5-s + (1.66e3 + 2.05e3i)6-s + 16.4i·7-s + (−1.10e4 − 3.53e3i)8-s − 6.01e3i·9-s + (−3.01e4 + 2.44e4i)10-s + (−3.59e4 + 3.59e4i)11-s + (−3.27e4 − 5.01e4i)12-s + (−7.03e4 − 7.03e4i)13-s + (38.3 − 369. i)14-s − 2.00e5·15-s + (2.40e5 + 1.05e5i)16-s − 4.91e5·17-s + ⋯
L(s)  = 1  + (−0.994 − 0.103i)2-s + (−0.589 − 0.589i)3-s + (0.978 + 0.205i)4-s + (0.868 − 0.868i)5-s + (0.525 + 0.647i)6-s + 0.00258i·7-s + (−0.952 − 0.305i)8-s − 0.305i·9-s + (−0.954 + 0.774i)10-s + (−0.740 + 0.740i)11-s + (−0.455 − 0.697i)12-s + (−0.683 − 0.683i)13-s + (0.000267 − 0.00257i)14-s − 1.02·15-s + (0.915 + 0.402i)16-s − 1.42·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0210i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.999 - 0.0210i$
Analytic conductor: \(8.24057\)
Root analytic conductor: \(2.87063\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :9/2),\ -0.999 - 0.0210i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.00476212 + 0.453383i\)
\(L(\frac12)\) \(\approx\) \(0.00476212 + 0.453383i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (22.5 + 2.33i)T \)
good3 \( 1 + (82.6 + 82.6i)T + 1.96e4iT^{2} \)
5 \( 1 + (-1.21e3 + 1.21e3i)T - 1.95e6iT^{2} \)
7 \( 1 - 16.4iT - 4.03e7T^{2} \)
11 \( 1 + (3.59e4 - 3.59e4i)T - 2.35e9iT^{2} \)
13 \( 1 + (7.03e4 + 7.03e4i)T + 1.06e10iT^{2} \)
17 \( 1 + 4.91e5T + 1.18e11T^{2} \)
19 \( 1 + (-4.40e5 - 4.40e5i)T + 3.22e11iT^{2} \)
23 \( 1 + 1.07e6iT - 1.80e12T^{2} \)
29 \( 1 + (1.85e6 + 1.85e6i)T + 1.45e13iT^{2} \)
31 \( 1 + 5.62e6T + 2.64e13T^{2} \)
37 \( 1 + (7.59e6 - 7.59e6i)T - 1.29e14iT^{2} \)
41 \( 1 + 2.84e7iT - 3.27e14T^{2} \)
43 \( 1 + (8.50e6 - 8.50e6i)T - 5.02e14iT^{2} \)
47 \( 1 - 3.94e7T + 1.11e15T^{2} \)
53 \( 1 + (-7.09e7 + 7.09e7i)T - 3.29e15iT^{2} \)
59 \( 1 + (5.24e7 - 5.24e7i)T - 8.66e15iT^{2} \)
61 \( 1 + (-3.32e7 - 3.32e7i)T + 1.16e16iT^{2} \)
67 \( 1 + (1.69e8 + 1.69e8i)T + 2.72e16iT^{2} \)
71 \( 1 - 1.00e6iT - 4.58e16T^{2} \)
73 \( 1 + 6.01e7iT - 5.88e16T^{2} \)
79 \( 1 - 2.97e8T + 1.19e17T^{2} \)
83 \( 1 + (5.17e8 + 5.17e8i)T + 1.86e17iT^{2} \)
89 \( 1 + 7.13e8iT - 3.50e17T^{2} \)
97 \( 1 - 7.49e8T + 7.60e17T^{2} \)
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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

−16.83439617755276405522770179307, −15.31135524527415776203251777830, −13.02664076615996140055831334174, −12.09965362394952832688924034730, −10.32812057642485487355795369109, −9.028128033446523345197749462854, −7.25338443585600762091305477514, −5.61274374472157096939796103394, −1.95640814821683176444583913979, −0.31469925132586321042976316386, 2.38176861741848182859315737847, 5.56888208274822576540792613065, 7.14134453497259075049344224395, 9.248938679985564609528123901075, 10.52965961770663247927370061034, 11.30459455991664311446705181410, 13.73035769192597264773656534057, 15.37190994273899986801338102847, 16.49880813997407529503446329864, 17.62083498432453413127029256849

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