Properties

Label 2-2e4-16.13-c13-0-14
Degree $2$
Conductor $16$
Sign $0.997 + 0.0667i$
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  + (−56.2 − 70.9i)2-s + (1.62e3 + 1.62e3i)3-s + (−1.86e3 + 7.97e3i)4-s + (8.03e3 − 8.03e3i)5-s + (2.37e4 − 2.06e5i)6-s − 5.16e5i·7-s + (6.70e5 − 3.16e5i)8-s + 3.66e6i·9-s + (−1.02e6 − 1.17e5i)10-s + (4.57e6 − 4.57e6i)11-s + (−1.59e7 + 9.91e6i)12-s + (1.23e7 + 1.23e7i)13-s + (−3.66e7 + 2.90e7i)14-s + 2.60e7·15-s + (−6.01e7 − 2.97e7i)16-s + 4.11e7·17-s + ⋯
L(s)  = 1  + (−0.621 − 0.783i)2-s + (1.28 + 1.28i)3-s + (−0.227 + 0.973i)4-s + (0.230 − 0.230i)5-s + (0.208 − 1.80i)6-s − 1.65i·7-s + (0.904 − 0.426i)8-s + 2.29i·9-s + (−0.323 − 0.0372i)10-s + (0.778 − 0.778i)11-s + (−1.54 + 0.958i)12-s + (0.707 + 0.707i)13-s + (−1.29 + 1.03i)14-s + 0.590·15-s + (−0.896 − 0.443i)16-s + 0.413·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0667i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.997 + 0.0667i$
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: \((2,\ 16,\ (\ :13/2),\ 0.997 + 0.0667i)\)

Particular Values

\(L(7)\) \(\approx\) \(2.32452 - 0.0776833i\)
\(L(\frac12)\) \(\approx\) \(2.32452 - 0.0776833i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (56.2 + 70.9i)T \)
good3 \( 1 + (-1.62e3 - 1.62e3i)T + 1.59e6iT^{2} \)
5 \( 1 + (-8.03e3 + 8.03e3i)T - 1.22e9iT^{2} \)
7 \( 1 + 5.16e5iT - 9.68e10T^{2} \)
11 \( 1 + (-4.57e6 + 4.57e6i)T - 3.45e13iT^{2} \)
13 \( 1 + (-1.23e7 - 1.23e7i)T + 3.02e14iT^{2} \)
17 \( 1 - 4.11e7T + 9.90e15T^{2} \)
19 \( 1 + (-2.09e8 - 2.09e8i)T + 4.20e16iT^{2} \)
23 \( 1 + 3.35e8iT - 5.04e17T^{2} \)
29 \( 1 + (1.25e9 + 1.25e9i)T + 1.02e19iT^{2} \)
31 \( 1 - 9.15e8T + 2.44e19T^{2} \)
37 \( 1 + (-9.04e9 + 9.04e9i)T - 2.43e20iT^{2} \)
41 \( 1 - 2.15e10iT - 9.25e20T^{2} \)
43 \( 1 + (-3.26e9 + 3.26e9i)T - 1.71e21iT^{2} \)
47 \( 1 - 7.63e10T + 5.46e21T^{2} \)
53 \( 1 + (-5.41e10 + 5.41e10i)T - 2.60e22iT^{2} \)
59 \( 1 + (-4.71e10 + 4.71e10i)T - 1.04e23iT^{2} \)
61 \( 1 + (3.10e11 + 3.10e11i)T + 1.61e23iT^{2} \)
67 \( 1 + (9.94e11 + 9.94e11i)T + 5.48e23iT^{2} \)
71 \( 1 - 1.34e12iT - 1.16e24T^{2} \)
73 \( 1 + 1.99e11iT - 1.67e24T^{2} \)
79 \( 1 + 1.43e12T + 4.66e24T^{2} \)
83 \( 1 + (2.07e10 + 2.07e10i)T + 8.87e24iT^{2} \)
89 \( 1 + 3.61e12iT - 2.19e25T^{2} \)
97 \( 1 + 4.73e12T + 6.73e25T^{2} \)
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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

−16.32097651751526809889820315525, −14.20176334521093122599186342616, −13.55381505378130778815573781217, −11.13321873493229775063456338063, −10.02570289556749282337640975928, −9.068654087113676332664416891793, −7.75102273278918028156712275511, −4.14308120799364302752445491742, −3.42129634382353350195806128556, −1.29999159577686646521544258841, 1.29383318829989463615642335733, 2.62743350357943623222375423233, 5.93407597314211805528768687699, 7.27131140858816118397082304905, 8.575334828218287391034745459981, 9.413109557491814498974438368818, 12.08127780428973325419873742920, 13.53895773969813305711906142830, 14.72427997359297086787983572932, 15.51859122396886430846017323746

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