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Properties

Label 2-2e4-16.3-c10-0-3
Degree $2$
Conductor $16$
Sign $-0.411 + 0.911i$
Analytic cond. $10.1657$
Root an. cond. $3.18837$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (17.5 + 26.7i)2-s + (−286. + 286. i)3-s + (−406. + 939. i)4-s + (−1.33e3 + 1.33e3i)5-s + (−1.26e4 − 2.62e3i)6-s + 1.96e4·7-s + (−3.22e4 + 5.63e3i)8-s − 1.04e5i·9-s + (−5.93e4 − 1.22e4i)10-s + (−1.92e4 − 1.92e4i)11-s + (−1.52e5 − 3.85e5i)12-s + (3.34e5 + 3.34e5i)13-s + (3.45e5 + 5.25e5i)14-s − 7.66e5i·15-s + (−7.17e5 − 7.64e5i)16-s − 2.63e6·17-s + ⋯
L(s)  = 1  + (0.549 + 0.835i)2-s + (−1.17 + 1.17i)3-s + (−0.397 + 0.917i)4-s + (−0.428 + 0.428i)5-s + (−1.63 − 0.337i)6-s + 1.16·7-s + (−0.985 + 0.172i)8-s − 1.77i·9-s + (−0.593 − 0.122i)10-s + (−0.119 − 0.119i)11-s + (−0.613 − 1.54i)12-s + (0.901 + 0.901i)13-s + (0.642 + 0.977i)14-s − 1.00i·15-s + (−0.684 − 0.728i)16-s − 1.85·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.411 + 0.911i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.411 + 0.911i$
Analytic conductor: \(10.1657\)
Root analytic conductor: \(3.18837\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :5),\ -0.411 + 0.911i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.505195 - 0.782261i\)
\(L(\frac12)\) \(\approx\) \(0.505195 - 0.782261i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-17.5 - 26.7i)T \)
good3 \( 1 + (286. - 286. i)T - 5.90e4iT^{2} \)
5 \( 1 + (1.33e3 - 1.33e3i)T - 9.76e6iT^{2} \)
7 \( 1 - 1.96e4T + 2.82e8T^{2} \)
11 \( 1 + (1.92e4 + 1.92e4i)T + 2.59e10iT^{2} \)
13 \( 1 + (-3.34e5 - 3.34e5i)T + 1.37e11iT^{2} \)
17 \( 1 + 2.63e6T + 2.01e12T^{2} \)
19 \( 1 + (-1.60e6 + 1.60e6i)T - 6.13e12iT^{2} \)
23 \( 1 + 4.82e6T + 4.14e13T^{2} \)
29 \( 1 + (-1.60e7 - 1.60e7i)T + 4.20e14iT^{2} \)
31 \( 1 - 1.59e6iT - 8.19e14T^{2} \)
37 \( 1 + (6.81e7 - 6.81e7i)T - 4.80e15iT^{2} \)
41 \( 1 + 1.62e8iT - 1.34e16T^{2} \)
43 \( 1 + (-3.72e7 - 3.72e7i)T + 2.16e16iT^{2} \)
47 \( 1 + 2.22e8iT - 5.25e16T^{2} \)
53 \( 1 + (2.79e8 - 2.79e8i)T - 1.74e17iT^{2} \)
59 \( 1 + (1.79e8 + 1.79e8i)T + 5.11e17iT^{2} \)
61 \( 1 + (-3.15e8 - 3.15e8i)T + 7.13e17iT^{2} \)
67 \( 1 + (1.34e9 - 1.34e9i)T - 1.82e18iT^{2} \)
71 \( 1 - 2.47e9T + 3.25e18T^{2} \)
73 \( 1 - 2.73e9iT - 4.29e18T^{2} \)
79 \( 1 - 4.24e8iT - 9.46e18T^{2} \)
83 \( 1 + (5.99e8 - 5.99e8i)T - 1.55e19iT^{2} \)
89 \( 1 - 7.29e9iT - 3.11e19T^{2} \)
97 \( 1 + 9.68e9T + 7.37e19T^{2} \)
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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

−17.31374878021159257117286022220, −15.95734562030405099995078287761, −15.35818636084567273752478071244, −13.89809273005117937335700082090, −11.70413292334482318993004722030, −10.99061189110437432309470925231, −8.792824984149820690097500590384, −6.72802885348095908848704981304, −5.11380183032100657055080621445, −4.04869017298966481831287615440, 0.43379015946050601699185731726, 1.74525358201027184786930268586, 4.67761542070545558659148498615, 6.09655331883611204706029639848, 8.146560820267241132964952109583, 10.81954663089864494222198967258, 11.66794583125394766214395137262, 12.68786437847077872406035638126, 13.83381646014896686292506096750, 15.73000000578307718306156454545

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