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Properties

Label 2-2e4-16.5-c13-0-23
Degree $2$
Conductor $16$
Sign $-0.381 + 0.924i$
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  + (83.6 − 34.6i)2-s + (1.37e3 − 1.37e3i)3-s + (5.78e3 − 5.79e3i)4-s + (−6.70e3 − 6.70e3i)5-s + (6.73e4 − 1.62e5i)6-s − 9.08e4i·7-s + (2.82e5 − 6.85e5i)8-s − 2.19e6i·9-s + (−7.92e5 − 3.28e5i)10-s + (3.10e6 + 3.10e6i)11-s + (−1.22e4 − 1.59e7i)12-s + (−1.61e7 + 1.61e7i)13-s + (−3.14e6 − 7.59e6i)14-s − 1.84e7·15-s + (−1.02e5 − 6.71e7i)16-s + 1.32e8·17-s + ⋯
L(s)  = 1  + (0.923 − 0.383i)2-s + (1.09 − 1.09i)3-s + (0.706 − 0.707i)4-s + (−0.191 − 0.191i)5-s + (0.589 − 1.42i)6-s − 0.291i·7-s + (0.381 − 0.924i)8-s − 1.37i·9-s + (−0.250 − 0.103i)10-s + (0.528 + 0.528i)11-s + (−0.00118 − 1.54i)12-s + (−0.930 + 0.930i)13-s + (−0.111 − 0.269i)14-s − 0.418·15-s + (−0.00153 − 0.999i)16-s + 1.32·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.381 + 0.924i$
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: \((2,\ 16,\ (\ :13/2),\ -0.381 + 0.924i)\)

Particular Values

\(L(7)\) \(\approx\) \(2.48430 - 3.71185i\)
\(L(\frac12)\) \(\approx\) \(2.48430 - 3.71185i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-83.6 + 34.6i)T \)
good3 \( 1 + (-1.37e3 + 1.37e3i)T - 1.59e6iT^{2} \)
5 \( 1 + (6.70e3 + 6.70e3i)T + 1.22e9iT^{2} \)
7 \( 1 + 9.08e4iT - 9.68e10T^{2} \)
11 \( 1 + (-3.10e6 - 3.10e6i)T + 3.45e13iT^{2} \)
13 \( 1 + (1.61e7 - 1.61e7i)T - 3.02e14iT^{2} \)
17 \( 1 - 1.32e8T + 9.90e15T^{2} \)
19 \( 1 + (1.57e8 - 1.57e8i)T - 4.20e16iT^{2} \)
23 \( 1 + 2.75e8iT - 5.04e17T^{2} \)
29 \( 1 + (-2.77e9 + 2.77e9i)T - 1.02e19iT^{2} \)
31 \( 1 + 5.11e9T + 2.44e19T^{2} \)
37 \( 1 + (-1.52e10 - 1.52e10i)T + 2.43e20iT^{2} \)
41 \( 1 - 4.02e10iT - 9.25e20T^{2} \)
43 \( 1 + (-3.13e10 - 3.13e10i)T + 1.71e21iT^{2} \)
47 \( 1 - 2.49e10T + 5.46e21T^{2} \)
53 \( 1 + (1.62e11 + 1.62e11i)T + 2.60e22iT^{2} \)
59 \( 1 + (-9.89e10 - 9.89e10i)T + 1.04e23iT^{2} \)
61 \( 1 + (2.79e11 - 2.79e11i)T - 1.61e23iT^{2} \)
67 \( 1 + (-8.56e11 + 8.56e11i)T - 5.48e23iT^{2} \)
71 \( 1 - 9.90e11iT - 1.16e24T^{2} \)
73 \( 1 - 1.09e12iT - 1.67e24T^{2} \)
79 \( 1 + 2.31e12T + 4.66e24T^{2} \)
83 \( 1 + (-7.01e11 + 7.01e11i)T - 8.87e24iT^{2} \)
89 \( 1 - 3.63e11iT - 2.19e25T^{2} \)
97 \( 1 + 1.33e13T + 6.73e25T^{2} \)
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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

−14.68091706972052279338566455844, −14.18955129457009717478449350301, −12.77307330291448112038306151034, −11.96744029644562065785232710799, −9.797073543779951371864339734536, −7.85380063702985114447846280919, −6.56582004061663662797064891442, −4.23562221532830984919450092271, −2.54858297441588321466574427440, −1.29207477502924841695935178334, 2.74202958743766592961166199517, 3.78589378049381951578370087338, 5.36492287881090881364527717048, 7.56928701694719331323875874396, 9.029113577216674644552190858085, 10.73026196949080983575312692443, 12.49269653255406548838007898992, 14.13403726822756681930864822368, 14.88969443062774297845193959968, 15.77969951001129034333510304055

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