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Properties

Label 2-2e4-16.13-c13-0-1
Degree $2$
Conductor $16$
Sign $-0.282 + 0.959i$
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  + (20.0 + 88.2i)2-s + (983. + 983. i)3-s + (−7.39e3 + 3.53e3i)4-s + (−3.53e4 + 3.53e4i)5-s + (−6.71e4 + 1.06e5i)6-s − 3.34e5i·7-s + (−4.59e5 − 5.81e5i)8-s + 3.40e5i·9-s + (−3.83e6 − 2.41e6i)10-s + (−3.18e6 + 3.18e6i)11-s + (−1.07e7 − 3.79e6i)12-s + (1.44e7 + 1.44e7i)13-s + (2.94e7 − 6.68e6i)14-s − 6.95e7·15-s + (4.21e7 − 5.22e7i)16-s − 2.55e7·17-s + ⋯
L(s)  = 1  + (0.221 + 0.975i)2-s + (0.778 + 0.778i)3-s + (−0.902 + 0.431i)4-s + (−1.01 + 1.01i)5-s + (−0.587 + 0.931i)6-s − 1.07i·7-s + (−0.620 − 0.784i)8-s + 0.213i·9-s + (−1.21 − 0.763i)10-s + (−0.542 + 0.542i)11-s + (−1.03 − 0.366i)12-s + (0.832 + 0.832i)13-s + (1.04 − 0.237i)14-s − 1.57·15-s + (0.628 − 0.778i)16-s − 0.256·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.282 + 0.959i$
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: \((2,\ 16,\ (\ :13/2),\ -0.282 + 0.959i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.505646 - 0.675958i\)
\(L(\frac12)\) \(\approx\) \(0.505646 - 0.675958i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-20.0 - 88.2i)T \)
good3 \( 1 + (-983. - 983. i)T + 1.59e6iT^{2} \)
5 \( 1 + (3.53e4 - 3.53e4i)T - 1.22e9iT^{2} \)
7 \( 1 + 3.34e5iT - 9.68e10T^{2} \)
11 \( 1 + (3.18e6 - 3.18e6i)T - 3.45e13iT^{2} \)
13 \( 1 + (-1.44e7 - 1.44e7i)T + 3.02e14iT^{2} \)
17 \( 1 + 2.55e7T + 9.90e15T^{2} \)
19 \( 1 + (1.03e8 + 1.03e8i)T + 4.20e16iT^{2} \)
23 \( 1 - 7.07e8iT - 5.04e17T^{2} \)
29 \( 1 + (2.65e9 + 2.65e9i)T + 1.02e19iT^{2} \)
31 \( 1 + 8.51e9T + 2.44e19T^{2} \)
37 \( 1 + (1.94e8 - 1.94e8i)T - 2.43e20iT^{2} \)
41 \( 1 - 4.34e10iT - 9.25e20T^{2} \)
43 \( 1 + (5.54e9 - 5.54e9i)T - 1.71e21iT^{2} \)
47 \( 1 + 1.27e11T + 5.46e21T^{2} \)
53 \( 1 + (2.22e11 - 2.22e11i)T - 2.60e22iT^{2} \)
59 \( 1 + (-2.40e11 + 2.40e11i)T - 1.04e23iT^{2} \)
61 \( 1 + (-3.98e11 - 3.98e11i)T + 1.61e23iT^{2} \)
67 \( 1 + (-1.28e11 - 1.28e11i)T + 5.48e23iT^{2} \)
71 \( 1 - 7.69e11iT - 1.16e24T^{2} \)
73 \( 1 + 1.77e12iT - 1.67e24T^{2} \)
79 \( 1 - 7.64e11T + 4.66e24T^{2} \)
83 \( 1 + (-1.62e12 - 1.62e12i)T + 8.87e24iT^{2} \)
89 \( 1 - 7.70e12iT - 2.19e25T^{2} \)
97 \( 1 + 9.45e12T + 6.73e25T^{2} \)
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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

−16.34560251102828135927153499939, −15.32483475507201160332383863874, −14.60530776583483486484568396479, −13.36186621434229192719774202777, −11.11511519981498312995161979608, −9.523364870593853532898987181890, −7.917158429501431539435016440232, −6.82225406871659814213169531578, −4.26336651932649954446973234349, −3.46792457314254730577079701052, 0.27084395654022301769486573055, 1.89500351241080704118687334904, 3.43786020075006960703003558389, 5.32920830133703402248613802208, 8.229437037709262873248685651585, 8.751008744300221123322745970126, 11.04507117448684329504578662314, 12.56288592704257898787564112945, 13.00651940918973847946404355683, 14.67814521491808752879422580011

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