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Properties

Label 2-2e4-16.5-c9-0-4
Degree $2$
Conductor $16$
Sign $-0.599 - 0.800i$
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  + (−3.01 + 22.4i)2-s + (80.3 − 80.3i)3-s + (−493. − 135. i)4-s + (652. + 652. i)5-s + (1.55e3 + 2.04e3i)6-s + 9.25e3i·7-s + (4.52e3 − 1.06e4i)8-s + 6.76e3i·9-s + (−1.65e4 + 1.26e4i)10-s + (−1.06e4 − 1.06e4i)11-s + (−5.05e4 + 2.88e4i)12-s + (−1.32e5 + 1.32e5i)13-s + (−2.07e5 − 2.79e4i)14-s + 1.04e5·15-s + (2.25e5 + 1.33e5i)16-s + 2.82e5·17-s + ⋯
L(s)  = 1  + (−0.133 + 0.991i)2-s + (0.572 − 0.572i)3-s + (−0.964 − 0.264i)4-s + (0.466 + 0.466i)5-s + (0.491 + 0.644i)6-s + 1.45i·7-s + (0.390 − 0.920i)8-s + 0.343i·9-s + (−0.524 + 0.400i)10-s + (−0.219 − 0.219i)11-s + (−0.703 + 0.401i)12-s + (−1.28 + 1.28i)13-s + (−1.44 − 0.194i)14-s + 0.534·15-s + (0.860 + 0.509i)16-s + 0.821·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.599 - 0.800i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.599 - 0.800i$
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: \((2,\ 16,\ (\ :9/2),\ -0.599 - 0.800i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.710791 + 1.42103i\)
\(L(\frac12)\) \(\approx\) \(0.710791 + 1.42103i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.01 - 22.4i)T \)
good3 \( 1 + (-80.3 + 80.3i)T - 1.96e4iT^{2} \)
5 \( 1 + (-652. - 652. i)T + 1.95e6iT^{2} \)
7 \( 1 - 9.25e3iT - 4.03e7T^{2} \)
11 \( 1 + (1.06e4 + 1.06e4i)T + 2.35e9iT^{2} \)
13 \( 1 + (1.32e5 - 1.32e5i)T - 1.06e10iT^{2} \)
17 \( 1 - 2.82e5T + 1.18e11T^{2} \)
19 \( 1 + (-1.37e5 + 1.37e5i)T - 3.22e11iT^{2} \)
23 \( 1 - 7.12e5iT - 1.80e12T^{2} \)
29 \( 1 + (-7.73e4 + 7.73e4i)T - 1.45e13iT^{2} \)
31 \( 1 - 7.32e6T + 2.64e13T^{2} \)
37 \( 1 + (5.72e6 + 5.72e6i)T + 1.29e14iT^{2} \)
41 \( 1 + 2.00e7iT - 3.27e14T^{2} \)
43 \( 1 + (1.31e7 + 1.31e7i)T + 5.02e14iT^{2} \)
47 \( 1 - 2.56e7T + 1.11e15T^{2} \)
53 \( 1 + (5.49e7 + 5.49e7i)T + 3.29e15iT^{2} \)
59 \( 1 + (-8.12e7 - 8.12e7i)T + 8.66e15iT^{2} \)
61 \( 1 + (-1.15e8 + 1.15e8i)T - 1.16e16iT^{2} \)
67 \( 1 + (1.94e7 - 1.94e7i)T - 2.72e16iT^{2} \)
71 \( 1 - 4.13e8iT - 4.58e16T^{2} \)
73 \( 1 - 2.83e8iT - 5.88e16T^{2} \)
79 \( 1 - 4.07e8T + 1.19e17T^{2} \)
83 \( 1 + (-1.88e8 + 1.88e8i)T - 1.86e17iT^{2} \)
89 \( 1 - 3.64e8iT - 3.50e17T^{2} \)
97 \( 1 + 1.37e9T + 7.60e17T^{2} \)
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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

−17.45600134755777344314441678862, −15.97503234338939535311947093202, −14.61547260035734608627924621239, −13.81670850754285484091861096497, −12.21980929932780216061667213520, −9.740277660220683230662804348245, −8.432798689285788545135702327080, −6.96219335208984208235252612305, −5.34332345090581389189592975718, −2.31489775354249542460033907480, 0.834264704995744189225654472019, 3.16116386792269009619199232250, 4.75591814527405178139887795300, 7.916764828178624158992597674736, 9.709822953468367350833100477709, 10.32231874753380468593007634446, 12.35960148787773006592319313800, 13.60146735734513114922164454082, 14.80125855931244872248988664369, 16.85592519936059804438663547370

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