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Properties

Label 2-2e4-16.3-c12-0-6
Degree $2$
Conductor $16$
Sign $0.999 + 0.0130i$
Analytic cond. $14.6239$
Root an. cond. $3.82412$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−18.3 − 61.3i)2-s + (37.8 − 37.8i)3-s + (−3.42e3 + 2.25e3i)4-s + (5.27e3 − 5.27e3i)5-s + (−3.01e3 − 1.62e3i)6-s − 1.15e5·7-s + (2.01e5 + 1.68e5i)8-s + 5.28e5i·9-s + (−4.19e5 − 2.26e5i)10-s + (2.08e6 + 2.08e6i)11-s + (−4.41e4 + 2.14e5i)12-s + (−4.74e6 − 4.74e6i)13-s + (2.11e6 + 7.06e6i)14-s − 3.98e5i·15-s + (6.62e6 − 1.54e7i)16-s + 4.57e7·17-s + ⋯
L(s)  = 1  + (−0.287 − 0.957i)2-s + (0.0518 − 0.0518i)3-s + (−0.835 + 0.550i)4-s + (0.337 − 0.337i)5-s + (−0.0646 − 0.0348i)6-s − 0.979·7-s + (0.766 + 0.641i)8-s + 0.994i·9-s + (−0.419 − 0.226i)10-s + (1.17 + 1.17i)11-s + (−0.0147 + 0.0718i)12-s + (−0.983 − 0.983i)13-s + (0.281 + 0.938i)14-s − 0.0350i·15-s + (0.394 − 0.918i)16-s + 1.89·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0130i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.999 + 0.0130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.999 + 0.0130i$
Analytic conductor: \(14.6239\)
Root analytic conductor: \(3.82412\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :6),\ 0.999 + 0.0130i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(1.30823 - 0.00854950i\)
\(L(\frac12)\) \(\approx\) \(1.30823 - 0.00854950i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (18.3 + 61.3i)T \)
good3 \( 1 + (-37.8 + 37.8i)T - 5.31e5iT^{2} \)
5 \( 1 + (-5.27e3 + 5.27e3i)T - 2.44e8iT^{2} \)
7 \( 1 + 1.15e5T + 1.38e10T^{2} \)
11 \( 1 + (-2.08e6 - 2.08e6i)T + 3.13e12iT^{2} \)
13 \( 1 + (4.74e6 + 4.74e6i)T + 2.32e13iT^{2} \)
17 \( 1 - 4.57e7T + 5.82e14T^{2} \)
19 \( 1 + (-1.18e6 + 1.18e6i)T - 2.21e15iT^{2} \)
23 \( 1 - 1.30e8T + 2.19e16T^{2} \)
29 \( 1 + (-8.44e7 - 8.44e7i)T + 3.53e17iT^{2} \)
31 \( 1 - 6.63e8iT - 7.87e17T^{2} \)
37 \( 1 + (1.56e9 - 1.56e9i)T - 6.58e18iT^{2} \)
41 \( 1 - 1.57e9iT - 2.25e19T^{2} \)
43 \( 1 + (-3.52e9 - 3.52e9i)T + 3.99e19iT^{2} \)
47 \( 1 - 1.56e10iT - 1.16e20T^{2} \)
53 \( 1 + (-9.93e9 + 9.93e9i)T - 4.91e20iT^{2} \)
59 \( 1 + (-2.74e10 - 2.74e10i)T + 1.77e21iT^{2} \)
61 \( 1 + (3.39e10 + 3.39e10i)T + 2.65e21iT^{2} \)
67 \( 1 + (6.51e10 - 6.51e10i)T - 8.18e21iT^{2} \)
71 \( 1 + 3.73e9T + 1.64e22T^{2} \)
73 \( 1 + 2.45e11iT - 2.29e22T^{2} \)
79 \( 1 - 3.43e11iT - 5.90e22T^{2} \)
83 \( 1 + (-3.73e11 + 3.73e11i)T - 1.06e23iT^{2} \)
89 \( 1 + 3.55e9iT - 2.46e23T^{2} \)
97 \( 1 - 2.84e11T + 6.93e23T^{2} \)
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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

−16.62538781984212992009429030531, −14.51987305087199300684777000490, −12.99522514131434395060499505448, −12.18217578278757197613314605249, −10.25808455036966717279600948449, −9.402941408517004997590601683014, −7.53149017590763880375191127541, −5.03270383498909408756067831105, −3.06651682441435860087085843784, −1.32398530963109561787286737353, 0.68318258721305446796736711197, 3.62128770269354830206184174360, 5.93770015213142271658996115810, 6.95641036164841109945187506369, 9.012176764212925383895220845669, 9.919387400160235492651407258206, 12.15001912410405366955563571919, 13.95474025352770194242103372426, 14.76428012149101546780439688972, 16.38886490727114041197204799804

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