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Properties

Label 2-2e4-16.5-c13-0-20
Degree $2$
Conductor $16$
Sign $-0.903 + 0.427i$
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  + (−18.7 − 88.5i)2-s + (1.61e3 − 1.61e3i)3-s + (−7.48e3 + 3.32e3i)4-s + (3.97e4 + 3.97e4i)5-s + (−1.72e5 − 1.12e5i)6-s − 4.39e5i·7-s + (4.34e5 + 6.00e5i)8-s − 3.59e6i·9-s + (2.77e6 − 4.26e6i)10-s + (−2.46e6 − 2.46e6i)11-s + (−6.71e6 + 1.74e7i)12-s + (5.83e6 − 5.83e6i)13-s + (−3.89e7 + 8.25e6i)14-s + 1.28e8·15-s + (4.50e7 − 4.97e7i)16-s − 6.96e7·17-s + ⋯
L(s)  = 1  + (−0.207 − 0.978i)2-s + (1.27 − 1.27i)3-s + (−0.914 + 0.405i)4-s + (1.13 + 1.13i)5-s + (−1.51 − 0.983i)6-s − 1.41i·7-s + (0.586 + 0.810i)8-s − 2.25i·9-s + (0.877 − 1.34i)10-s + (−0.418 − 0.418i)11-s + (−0.648 + 1.68i)12-s + (0.335 − 0.335i)13-s + (−1.38 + 0.292i)14-s + 2.90·15-s + (0.671 − 0.741i)16-s − 0.699·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 + 0.427i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.903 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.903 + 0.427i$
Analytic conductor: \(17.1569\)
Root analytic conductor: \(4.14209\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :13/2),\ -0.903 + 0.427i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.585794 - 2.60651i\)
\(L(\frac12)\) \(\approx\) \(0.585794 - 2.60651i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (18.7 + 88.5i)T \)
good3 \( 1 + (-1.61e3 + 1.61e3i)T - 1.59e6iT^{2} \)
5 \( 1 + (-3.97e4 - 3.97e4i)T + 1.22e9iT^{2} \)
7 \( 1 + 4.39e5iT - 9.68e10T^{2} \)
11 \( 1 + (2.46e6 + 2.46e6i)T + 3.45e13iT^{2} \)
13 \( 1 + (-5.83e6 + 5.83e6i)T - 3.02e14iT^{2} \)
17 \( 1 + 6.96e7T + 9.90e15T^{2} \)
19 \( 1 + (-1.89e7 + 1.89e7i)T - 4.20e16iT^{2} \)
23 \( 1 - 1.01e8iT - 5.04e17T^{2} \)
29 \( 1 + (1.53e9 - 1.53e9i)T - 1.02e19iT^{2} \)
31 \( 1 - 6.07e9T + 2.44e19T^{2} \)
37 \( 1 + (-2.43e9 - 2.43e9i)T + 2.43e20iT^{2} \)
41 \( 1 - 2.21e10iT - 9.25e20T^{2} \)
43 \( 1 + (-1.85e10 - 1.85e10i)T + 1.71e21iT^{2} \)
47 \( 1 + 2.26e9T + 5.46e21T^{2} \)
53 \( 1 + (1.71e11 + 1.71e11i)T + 2.60e22iT^{2} \)
59 \( 1 + (-3.32e11 - 3.32e11i)T + 1.04e23iT^{2} \)
61 \( 1 + (-4.18e11 + 4.18e11i)T - 1.61e23iT^{2} \)
67 \( 1 + (-6.22e11 + 6.22e11i)T - 5.48e23iT^{2} \)
71 \( 1 + 4.34e10iT - 1.16e24T^{2} \)
73 \( 1 + 2.30e11iT - 1.67e24T^{2} \)
79 \( 1 - 2.02e12T + 4.66e24T^{2} \)
83 \( 1 + (3.13e12 - 3.13e12i)T - 8.87e24iT^{2} \)
89 \( 1 - 7.10e12iT - 2.19e25T^{2} \)
97 \( 1 - 9.82e12T + 6.73e25T^{2} \)
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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

−14.44765658859117414066697070604, −13.63516443146302679167703413834, −13.12884557275758904426412852946, −10.93938671111765481245384864982, −9.702799430967870930328380622828, −8.051381140566029871095645891192, −6.73426879274475180370985477545, −3.38229605973539591907652936132, −2.30713427255887149370509730754, −1.00631625933120212452035926162, 2.18225578636112610843740497351, 4.54099843053566500382740039422, 5.63463854462965506230417200571, 8.440237242790411343919336708351, 9.084314769034269110193841456978, 9.927380792743339934006008722689, 13.03377972402921761630207158294, 14.13579605556104841852091162155, 15.38786093829813581031452651972, 16.05121056883179399621266296990

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