Properties

Label 2-2e4-16.5-c13-0-21
Degree $2$
Conductor $16$
Sign $-0.999 - 0.0424i$
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  + (−90.1 + 7.91i)2-s + (1.17e3 − 1.17e3i)3-s + (8.06e3 − 1.42e3i)4-s + (−3.31e4 − 3.31e4i)5-s + (−9.66e4 + 1.15e5i)6-s − 5.48e5i·7-s + (−7.16e5 + 1.92e5i)8-s − 1.16e6i·9-s + (3.24e6 + 2.72e6i)10-s + (4.11e6 + 4.11e6i)11-s + (7.79e6 − 1.11e7i)12-s + (1.23e7 − 1.23e7i)13-s + (4.34e6 + 4.94e7i)14-s − 7.77e7·15-s + (6.30e7 − 2.30e7i)16-s − 1.10e8·17-s + ⋯
L(s)  = 1  + (−0.996 + 0.0874i)2-s + (0.930 − 0.930i)3-s + (0.984 − 0.174i)4-s + (−0.947 − 0.947i)5-s + (−0.845 + 1.00i)6-s − 1.76i·7-s + (−0.965 + 0.259i)8-s − 0.730i·9-s + (1.02 + 0.861i)10-s + (0.700 + 0.700i)11-s + (0.753 − 1.07i)12-s + (0.707 − 0.707i)13-s + (0.154 + 1.75i)14-s − 1.76·15-s + (0.939 − 0.343i)16-s − 1.11·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0424i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.999 - 0.0424i$
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: \((2,\ 16,\ (\ :13/2),\ -0.999 - 0.0424i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.0222733 + 1.04893i\)
\(L(\frac12)\) \(\approx\) \(0.0222733 + 1.04893i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (90.1 - 7.91i)T \)
good3 \( 1 + (-1.17e3 + 1.17e3i)T - 1.59e6iT^{2} \)
5 \( 1 + (3.31e4 + 3.31e4i)T + 1.22e9iT^{2} \)
7 \( 1 + 5.48e5iT - 9.68e10T^{2} \)
11 \( 1 + (-4.11e6 - 4.11e6i)T + 3.45e13iT^{2} \)
13 \( 1 + (-1.23e7 + 1.23e7i)T - 3.02e14iT^{2} \)
17 \( 1 + 1.10e8T + 9.90e15T^{2} \)
19 \( 1 + (2.56e7 - 2.56e7i)T - 4.20e16iT^{2} \)
23 \( 1 - 6.87e8iT - 5.04e17T^{2} \)
29 \( 1 + (-1.49e9 + 1.49e9i)T - 1.02e19iT^{2} \)
31 \( 1 + 7.17e9T + 2.44e19T^{2} \)
37 \( 1 + (1.12e10 + 1.12e10i)T + 2.43e20iT^{2} \)
41 \( 1 + 1.09e10iT - 9.25e20T^{2} \)
43 \( 1 + (-1.78e9 - 1.78e9i)T + 1.71e21iT^{2} \)
47 \( 1 - 9.44e10T + 5.46e21T^{2} \)
53 \( 1 + (-1.79e11 - 1.79e11i)T + 2.60e22iT^{2} \)
59 \( 1 + (2.67e11 + 2.67e11i)T + 1.04e23iT^{2} \)
61 \( 1 + (1.55e11 - 1.55e11i)T - 1.61e23iT^{2} \)
67 \( 1 + (-4.23e11 + 4.23e11i)T - 5.48e23iT^{2} \)
71 \( 1 + 7.99e11iT - 1.16e24T^{2} \)
73 \( 1 - 3.27e11iT - 1.67e24T^{2} \)
79 \( 1 - 3.12e12T + 4.66e24T^{2} \)
83 \( 1 + (1.35e12 - 1.35e12i)T - 8.87e24iT^{2} \)
89 \( 1 - 4.52e11iT - 2.19e25T^{2} \)
97 \( 1 - 5.86e12T + 6.73e25T^{2} \)
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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

−15.44978851006892755523975138289, −13.78296121803539603324134002026, −12.50837919555277967144259175762, −10.85297256314734282901787260752, −9.001900793281367271553644889876, −7.83207577011468616055791623413, −7.08083977754882188310715422082, −3.85413189023763093493233916024, −1.55765231022775678544919746477, −0.50549577404448809975549294255, 2.43850643399324192011613266509, 3.60476008324062587536952987557, 6.53204359840835929084218691239, 8.584066531169464514206801950768, 9.035414464227801840810448724646, 10.86649976751773423115800264737, 11.88067401955022004402440013057, 14.62057654547612493521013788878, 15.38481518038363750326101176211, 16.16484420831558734478904987574

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