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Properties

Label 2-2e4-16.3-c12-0-18
Degree $2$
Conductor $16$
Sign $-0.969 + 0.246i$
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  + (26.6 − 58.2i)2-s + (−435. + 435. i)3-s + (−2.68e3 − 3.09e3i)4-s + (1.73e4 − 1.73e4i)5-s + (1.37e4 + 3.69e4i)6-s + 1.02e5·7-s + (−2.51e5 + 7.36e4i)8-s + 1.51e5i·9-s + (−5.47e5 − 1.47e6i)10-s + (−1.38e6 − 1.38e6i)11-s + (2.51e6 + 1.81e5i)12-s + (−5.77e6 − 5.77e6i)13-s + (2.72e6 − 5.96e6i)14-s + 1.51e7i·15-s + (−2.40e6 + 1.66e7i)16-s − 3.78e7·17-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)2-s + (−0.597 + 0.597i)3-s + (−0.654 − 0.756i)4-s + (1.10 − 1.10i)5-s + (0.295 + 0.791i)6-s + 0.870·7-s + (−0.959 + 0.280i)8-s + 0.285i·9-s + (−0.547 − 1.47i)10-s + (−0.781 − 0.781i)11-s + (0.842 + 0.0608i)12-s + (−1.19 − 1.19i)13-s + (0.361 − 0.791i)14-s + 1.32i·15-s + (−0.143 + 0.989i)16-s − 1.56·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.969 + 0.246i$
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: \((2,\ 16,\ (\ :6),\ -0.969 + 0.246i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.183094 - 1.46554i\)
\(L(\frac12)\) \(\approx\) \(0.183094 - 1.46554i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-26.6 + 58.2i)T \)
good3 \( 1 + (435. - 435. i)T - 5.31e5iT^{2} \)
5 \( 1 + (-1.73e4 + 1.73e4i)T - 2.44e8iT^{2} \)
7 \( 1 - 1.02e5T + 1.38e10T^{2} \)
11 \( 1 + (1.38e6 + 1.38e6i)T + 3.13e12iT^{2} \)
13 \( 1 + (5.77e6 + 5.77e6i)T + 2.32e13iT^{2} \)
17 \( 1 + 3.78e7T + 5.82e14T^{2} \)
19 \( 1 + (-2.52e7 + 2.52e7i)T - 2.21e15iT^{2} \)
23 \( 1 - 1.57e8T + 2.19e16T^{2} \)
29 \( 1 + (-5.46e7 - 5.46e7i)T + 3.53e17iT^{2} \)
31 \( 1 - 6.13e8iT - 7.87e17T^{2} \)
37 \( 1 + (5.91e8 - 5.91e8i)T - 6.58e18iT^{2} \)
41 \( 1 + 6.47e9iT - 2.25e19T^{2} \)
43 \( 1 + (6.63e8 + 6.63e8i)T + 3.99e19iT^{2} \)
47 \( 1 + 6.80e9iT - 1.16e20T^{2} \)
53 \( 1 + (-1.10e10 + 1.10e10i)T - 4.91e20iT^{2} \)
59 \( 1 + (-1.35e10 - 1.35e10i)T + 1.77e21iT^{2} \)
61 \( 1 + (-2.49e10 - 2.49e10i)T + 2.65e21iT^{2} \)
67 \( 1 + (-6.49e10 + 6.49e10i)T - 8.18e21iT^{2} \)
71 \( 1 - 2.09e11T + 1.64e22T^{2} \)
73 \( 1 - 9.47e10iT - 2.29e22T^{2} \)
79 \( 1 - 6.54e10iT - 5.90e22T^{2} \)
83 \( 1 + (-5.21e10 + 5.21e10i)T - 1.06e23iT^{2} \)
89 \( 1 - 1.94e11iT - 2.46e23T^{2} \)
97 \( 1 + 3.18e11T + 6.93e23T^{2} \)
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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

−15.49071927901642341716619887430, −13.74456196353356460102381989752, −12.82946086149054744502344041959, −11.17585121191122721639863516039, −10.16820106511954225292126339587, −8.712904417135228232886554160406, −5.23089904639724023129736549971, −5.03456596528327914442926063625, −2.29431692853502689017981135922, −0.54202398499202077050686926239, 2.23707344191191172658012361079, 4.91343827711329071610724399269, 6.44521455395228044482504349268, 7.30836891917389494096241799978, 9.539470047822066267887960777991, 11.44163250915748094597511505027, 12.96750840471117276000807398714, 14.25251304183786580440424804042, 15.10881696479929160616516735156, 17.13885785715493045991334705060

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