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Properties

Label 2-2e4-16.13-c13-0-9
Degree $2$
Conductor $16$
Sign $0.916 - 0.399i$
Analytic cond. $17.1569$
Root an. cond. $4.14209$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (64.2 + 63.7i)2-s + (−1.61e3 − 1.61e3i)3-s + (73.4 + 8.19e3i)4-s + (−2.31e4 + 2.31e4i)5-s + (−928. − 2.07e5i)6-s − 2.80e5i·7-s + (−5.17e5 + 5.31e5i)8-s + 3.64e6i·9-s + (−2.96e6 + 1.33e4i)10-s + (6.11e6 − 6.11e6i)11-s + (1.31e7 − 1.33e7i)12-s + (1.15e7 + 1.15e7i)13-s + (1.78e7 − 1.80e7i)14-s + 7.50e7·15-s + (−6.70e7 + 1.20e6i)16-s + 1.68e8·17-s + ⋯
L(s)  = 1  + (0.710 + 0.703i)2-s + (−1.28 − 1.28i)3-s + (0.00896 + 0.999i)4-s + (−0.663 + 0.663i)5-s + (−0.00812 − 1.81i)6-s − 0.900i·7-s + (−0.697 + 0.716i)8-s + 2.28i·9-s + (−0.938 + 0.00420i)10-s + (1.04 − 1.04i)11-s + (1.27 − 1.29i)12-s + (0.662 + 0.662i)13-s + (0.633 − 0.639i)14-s + 1.70·15-s + (−0.999 + 0.0179i)16-s + 1.69·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.916 - 0.399i$
Analytic conductor: \(17.1569\)
Root analytic conductor: \(4.14209\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :13/2),\ 0.916 - 0.399i)\)

Particular Values

\(L(7)\) \(\approx\) \(1.54726 + 0.322221i\)
\(L(\frac12)\) \(\approx\) \(1.54726 + 0.322221i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-64.2 - 63.7i)T \)
good3 \( 1 + (1.61e3 + 1.61e3i)T + 1.59e6iT^{2} \)
5 \( 1 + (2.31e4 - 2.31e4i)T - 1.22e9iT^{2} \)
7 \( 1 + 2.80e5iT - 9.68e10T^{2} \)
11 \( 1 + (-6.11e6 + 6.11e6i)T - 3.45e13iT^{2} \)
13 \( 1 + (-1.15e7 - 1.15e7i)T + 3.02e14iT^{2} \)
17 \( 1 - 1.68e8T + 9.90e15T^{2} \)
19 \( 1 + (-5.82e6 - 5.82e6i)T + 4.20e16iT^{2} \)
23 \( 1 - 2.84e8iT - 5.04e17T^{2} \)
29 \( 1 + (1.26e9 + 1.26e9i)T + 1.02e19iT^{2} \)
31 \( 1 - 1.42e9T + 2.44e19T^{2} \)
37 \( 1 + (-1.30e10 + 1.30e10i)T - 2.43e20iT^{2} \)
41 \( 1 - 3.32e10iT - 9.25e20T^{2} \)
43 \( 1 + (8.67e9 - 8.67e9i)T - 1.71e21iT^{2} \)
47 \( 1 - 2.33e10T + 5.46e21T^{2} \)
53 \( 1 + (-2.15e11 + 2.15e11i)T - 2.60e22iT^{2} \)
59 \( 1 + (1.81e11 - 1.81e11i)T - 1.04e23iT^{2} \)
61 \( 1 + (1.87e10 + 1.87e10i)T + 1.61e23iT^{2} \)
67 \( 1 + (-1.72e11 - 1.72e11i)T + 5.48e23iT^{2} \)
71 \( 1 - 3.76e11iT - 1.16e24T^{2} \)
73 \( 1 - 1.75e12iT - 1.67e24T^{2} \)
79 \( 1 - 3.85e12T + 4.66e24T^{2} \)
83 \( 1 + (8.88e10 + 8.88e10i)T + 8.87e24iT^{2} \)
89 \( 1 - 1.77e12iT - 2.19e25T^{2} \)
97 \( 1 - 1.04e13T + 6.73e25T^{2} \)
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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

−16.42665170554588279082879342809, −14.34614946562332494021655201747, −13.34817571468706117424630971840, −11.85337663315934522133001608674, −11.20509646869735305149479180239, −7.82424680297038663859164868059, −6.85899919151216601469620914678, −5.82478220464472553898860513089, −3.73457787901477738818583494695, −0.944072873906281265096294317529, 0.855665529096952878239068411543, 3.70946561668058666609162131533, 4.86370226053088446080559772969, 5.96188587945319773413642737105, 9.251086947245754548176116158601, 10.45936440560962355911416380517, 11.97199789474685693394270644485, 12.23829224635881357265781016545, 14.84692017889045348320222853630, 15.65728635323634411572680182561

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