Properties

Label 2-2e4-16.5-c9-0-9
Degree $2$
Conductor $16$
Sign $0.943 - 0.331i$
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  + (−12.8 + 18.6i)2-s + (−4.54 + 4.54i)3-s + (−182. − 478. i)4-s + (335. + 335. i)5-s + (−26.4 − 142. i)6-s − 1.15e4i·7-s + (1.12e4 + 2.72e3i)8-s + 1.96e4i·9-s + (−1.05e4 + 1.95e3i)10-s + (5.62e4 + 5.62e4i)11-s + (3.00e3 + 1.34e3i)12-s + (8.90e4 − 8.90e4i)13-s + (2.15e5 + 1.47e5i)14-s − 3.04e3·15-s + (−1.95e5 + 1.74e5i)16-s + 5.74e4·17-s + ⋯
L(s)  = 1  + (−0.566 + 0.823i)2-s + (−0.0323 + 0.0323i)3-s + (−0.357 − 0.933i)4-s + (0.240 + 0.240i)5-s + (−0.00832 − 0.0450i)6-s − 1.81i·7-s + (0.971 + 0.235i)8-s + 0.997i·9-s + (−0.333 + 0.0616i)10-s + (1.15 + 1.15i)11-s + (0.0418 + 0.0186i)12-s + (0.864 − 0.864i)13-s + (1.49 + 1.02i)14-s − 0.0155·15-s + (−0.744 + 0.667i)16-s + 0.166·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.943 - 0.331i$
Analytic conductor: \(8.24057\)
Root analytic conductor: \(2.87063\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :9/2),\ 0.943 - 0.331i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.32351 + 0.225874i\)
\(L(\frac12)\) \(\approx\) \(1.32351 + 0.225874i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (12.8 - 18.6i)T \)
good3 \( 1 + (4.54 - 4.54i)T - 1.96e4iT^{2} \)
5 \( 1 + (-335. - 335. i)T + 1.95e6iT^{2} \)
7 \( 1 + 1.15e4iT - 4.03e7T^{2} \)
11 \( 1 + (-5.62e4 - 5.62e4i)T + 2.35e9iT^{2} \)
13 \( 1 + (-8.90e4 + 8.90e4i)T - 1.06e10iT^{2} \)
17 \( 1 - 5.74e4T + 1.18e11T^{2} \)
19 \( 1 + (-4.64e5 + 4.64e5i)T - 3.22e11iT^{2} \)
23 \( 1 + 4.00e5iT - 1.80e12T^{2} \)
29 \( 1 + (2.95e5 - 2.95e5i)T - 1.45e13iT^{2} \)
31 \( 1 - 7.36e6T + 2.64e13T^{2} \)
37 \( 1 + (5.69e6 + 5.69e6i)T + 1.29e14iT^{2} \)
41 \( 1 - 1.77e7iT - 3.27e14T^{2} \)
43 \( 1 + (-1.40e7 - 1.40e7i)T + 5.02e14iT^{2} \)
47 \( 1 - 7.65e6T + 1.11e15T^{2} \)
53 \( 1 + (3.32e7 + 3.32e7i)T + 3.29e15iT^{2} \)
59 \( 1 + (5.24e7 + 5.24e7i)T + 8.66e15iT^{2} \)
61 \( 1 + (7.40e7 - 7.40e7i)T - 1.16e16iT^{2} \)
67 \( 1 + (-5.58e7 + 5.58e7i)T - 2.72e16iT^{2} \)
71 \( 1 - 4.59e7iT - 4.58e16T^{2} \)
73 \( 1 - 1.31e8iT - 5.88e16T^{2} \)
79 \( 1 - 2.74e7T + 1.19e17T^{2} \)
83 \( 1 + (3.58e7 - 3.58e7i)T - 1.86e17iT^{2} \)
89 \( 1 - 7.45e8iT - 3.50e17T^{2} \)
97 \( 1 - 9.87e8T + 7.60e17T^{2} \)
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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

−17.10112986296427135125538040447, −15.94566204468628024059995604173, −14.32450878720451207822833954516, −13.46104751359472312062516303144, −10.83255770568260128449225553731, −9.859268351479380808420283179099, −7.84785666878626208533213999272, −6.68807973739053198728607264486, −4.52212677383918450962884421600, −1.07086354103825508212518354503, 1.36878671471515516618752860231, 3.40700541477382297116039187469, 6.04919537689707304239350224774, 8.722545751153317779713928448399, 9.318859254686992546171572111090, 11.56941482549605273134157769784, 12.17312494318797679577311632173, 13.94747610758801748716179123764, 15.76003125803560483666818231120, 17.15265337151274673276334168635

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