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Properties

Label 2-2e4-16.13-c21-0-34
Degree $2$
Conductor $16$
Sign $0.658 - 0.752i$
Analytic cond. $44.7163$
Root an. cond. $6.68703$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.13e3 − 899. i)2-s + (−1.27e5 − 1.27e5i)3-s + (4.77e5 + 2.04e6i)4-s + (6.46e6 − 6.46e6i)5-s + (2.99e7 + 2.59e8i)6-s − 1.09e9i·7-s + (1.29e9 − 2.74e9i)8-s + 2.21e10i·9-s + (−1.31e10 + 1.51e9i)10-s + (9.04e10 − 9.04e10i)11-s + (1.99e11 − 3.21e11i)12-s + (6.65e10 + 6.65e10i)13-s + (−9.83e11 + 1.24e12i)14-s − 1.65e12·15-s + (−3.94e12 + 1.95e12i)16-s − 1.23e13·17-s + ⋯
L(s)  = 1  + (−0.783 − 0.621i)2-s + (−1.24 − 1.24i)3-s + (0.227 + 0.973i)4-s + (0.296 − 0.296i)5-s + (0.202 + 1.75i)6-s − 1.46i·7-s + (0.426 − 0.904i)8-s + 2.11i·9-s + (−0.416 + 0.0480i)10-s + (1.05 − 1.05i)11-s + (0.931 − 1.49i)12-s + (0.133 + 0.133i)13-s + (−0.909 + 1.14i)14-s − 0.739·15-s + (−0.896 + 0.443i)16-s − 1.48·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.658 - 0.752i$
Analytic conductor: \(44.7163\)
Root analytic conductor: \(6.68703\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :21/2),\ 0.658 - 0.752i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.3846245060\)
\(L(\frac12)\) \(\approx\) \(0.3846245060\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13e3 + 899. i)T \)
good3 \( 1 + (1.27e5 + 1.27e5i)T + 1.04e10iT^{2} \)
5 \( 1 + (-6.46e6 + 6.46e6i)T - 4.76e14iT^{2} \)
7 \( 1 + 1.09e9iT - 5.58e17T^{2} \)
11 \( 1 + (-9.04e10 + 9.04e10i)T - 7.40e21iT^{2} \)
13 \( 1 + (-6.65e10 - 6.65e10i)T + 2.47e23iT^{2} \)
17 \( 1 + 1.23e13T + 6.90e25T^{2} \)
19 \( 1 + (2.96e13 + 2.96e13i)T + 7.14e26iT^{2} \)
23 \( 1 + 1.97e14iT - 3.94e28T^{2} \)
29 \( 1 + (9.11e14 + 9.11e14i)T + 5.13e30iT^{2} \)
31 \( 1 + 4.52e15T + 2.08e31T^{2} \)
37 \( 1 + (-2.37e16 + 2.37e16i)T - 8.55e32iT^{2} \)
41 \( 1 + 4.95e16iT - 7.38e33T^{2} \)
43 \( 1 + (5.80e16 - 5.80e16i)T - 2.00e34iT^{2} \)
47 \( 1 - 1.35e17T + 1.30e35T^{2} \)
53 \( 1 + (-1.22e18 + 1.22e18i)T - 1.62e36iT^{2} \)
59 \( 1 + (2.88e18 - 2.88e18i)T - 1.54e37iT^{2} \)
61 \( 1 + (2.51e18 + 2.51e18i)T + 3.10e37iT^{2} \)
67 \( 1 + (-1.76e18 - 1.76e18i)T + 2.22e38iT^{2} \)
71 \( 1 + 6.89e18iT - 7.52e38T^{2} \)
73 \( 1 - 2.30e19iT - 1.34e39T^{2} \)
79 \( 1 - 5.60e18T + 7.08e39T^{2} \)
83 \( 1 + (5.63e18 + 5.63e18i)T + 1.99e40iT^{2} \)
89 \( 1 + 1.52e20iT - 8.65e40T^{2} \)
97 \( 1 + 2.96e20T + 5.27e41T^{2} \)
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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

−12.88805221478408275643088908684, −11.28661859544437246456311776753, −10.86574597463444154796176019427, −8.830447473112981318081041985921, −7.19507138021606224534699266252, −6.40611124830608148724066464463, −4.21948493327685148247751721482, −1.94986080605915100177666464063, −0.807668518343274611649476548381, −0.22766244630423451261853537239, 1.88417266115006295760446711685, 4.42071316443535726118913902885, 5.73061780111199363125035259329, 6.51631773585836028081495779822, 8.912704706836829935581976530642, 9.774013098755618045330871964053, 10.97921624986748022368762441075, 12.09417436648758136096224783936, 14.89304130186253330864002591016, 15.35794926041606987651064790123

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