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Properties

Label 2-2e4-16.13-c5-0-3
Degree $2$
Conductor $16$
Sign $0.575 - 0.817i$
Analytic cond. $2.56614$
Root an. cond. $1.60191$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.52 + 3.39i)2-s + (3.84 + 3.84i)3-s + (8.91 + 30.7i)4-s + (8.73 − 8.73i)5-s + (4.32 + 30.4i)6-s − 28.0i·7-s + (−64.0 + 169. i)8-s − 213. i·9-s + (69.1 − 9.83i)10-s + (191. − 191. i)11-s + (−83.7 + 152. i)12-s + (−562. − 562. i)13-s + (95.1 − 126. i)14-s + 67.0·15-s + (−864. + 548. i)16-s − 663.·17-s + ⋯
L(s)  = 1  + (0.799 + 0.600i)2-s + (0.246 + 0.246i)3-s + (0.278 + 0.960i)4-s + (0.156 − 0.156i)5-s + (0.0490 + 0.344i)6-s − 0.216i·7-s + (−0.353 + 0.935i)8-s − 0.878i·9-s + (0.218 − 0.0310i)10-s + (0.478 − 0.478i)11-s + (−0.167 + 0.305i)12-s + (−0.923 − 0.923i)13-s + (0.129 − 0.172i)14-s + 0.0769·15-s + (−0.844 + 0.535i)16-s − 0.556·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.575 - 0.817i$
Analytic conductor: \(2.56614\)
Root analytic conductor: \(1.60191\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :5/2),\ 0.575 - 0.817i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.81539 + 0.942220i\)
\(L(\frac12)\) \(\approx\) \(1.81539 + 0.942220i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.52 - 3.39i)T \)
good3 \( 1 + (-3.84 - 3.84i)T + 243iT^{2} \)
5 \( 1 + (-8.73 + 8.73i)T - 3.12e3iT^{2} \)
7 \( 1 + 28.0iT - 1.68e4T^{2} \)
11 \( 1 + (-191. + 191. i)T - 1.61e5iT^{2} \)
13 \( 1 + (562. + 562. i)T + 3.71e5iT^{2} \)
17 \( 1 + 663.T + 1.41e6T^{2} \)
19 \( 1 + (-1.79e3 - 1.79e3i)T + 2.47e6iT^{2} \)
23 \( 1 - 2.87e3iT - 6.43e6T^{2} \)
29 \( 1 + (3.90e3 + 3.90e3i)T + 2.05e7iT^{2} \)
31 \( 1 + 4.45e3T + 2.86e7T^{2} \)
37 \( 1 + (-6.04e3 + 6.04e3i)T - 6.93e7iT^{2} \)
41 \( 1 - 1.23e4iT - 1.15e8T^{2} \)
43 \( 1 + (-1.06e4 + 1.06e4i)T - 1.47e8iT^{2} \)
47 \( 1 + 2.33e4T + 2.29e8T^{2} \)
53 \( 1 + (-656. + 656. i)T - 4.18e8iT^{2} \)
59 \( 1 + (5.98e3 - 5.98e3i)T - 7.14e8iT^{2} \)
61 \( 1 + (-6.86e3 - 6.86e3i)T + 8.44e8iT^{2} \)
67 \( 1 + (-1.30e4 - 1.30e4i)T + 1.35e9iT^{2} \)
71 \( 1 - 2.78e4iT - 1.80e9T^{2} \)
73 \( 1 + 2.05e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.03e5T + 3.07e9T^{2} \)
83 \( 1 + (8.21e3 + 8.21e3i)T + 3.93e9iT^{2} \)
89 \( 1 + 1.34e5iT - 5.58e9T^{2} \)
97 \( 1 + 5.36e4T + 8.58e9T^{2} \)
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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

−17.88036021017113081091870380565, −16.77066574634551574105094473980, −15.38890967772783944243525848241, −14.40107262901416124098678018191, −13.02997206999193602239689638178, −11.65163500897864035101276540443, −9.416664209818423471163577314010, −7.56691221893669486191551305490, −5.68319862056063564988512332735, −3.58500562024181278648616376900, 2.29676505874057226414737492002, 4.78977856118811118937165375806, 6.93831138581450350903403264858, 9.417585922980827082852968970045, 11.06306573572464482219799775004, 12.43421134719182089760981816811, 13.77958324286759359845116667114, 14.75436776557654253463404667777, 16.35565806693056024049140479598, 18.23410551745229222828906500391

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