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Properties

Label 2-2e4-16.13-c21-0-39
Degree $2$
Conductor $16$
Sign $0.202 - 0.979i$
Analytic cond. $44.7163$
Root an. cond. $6.68703$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (746. − 1.24e3i)2-s + (−2.29e4 − 2.29e4i)3-s + (−9.81e5 − 1.85e6i)4-s + (−9.16e6 + 9.16e6i)5-s + (−4.56e7 + 1.13e7i)6-s − 5.91e8i·7-s + (−3.03e9 − 1.66e8i)8-s − 9.40e9i·9-s + (4.52e9 + 1.82e10i)10-s + (4.03e10 − 4.03e10i)11-s + (−2.00e10 + 6.51e10i)12-s + (−4.34e11 − 4.34e11i)13-s + (−7.33e11 − 4.41e11i)14-s + 4.21e11·15-s + (−2.47e12 + 3.63e12i)16-s − 3.88e12·17-s + ⋯
L(s)  = 1  + (0.515 − 0.856i)2-s + (−0.224 − 0.224i)3-s + (−0.467 − 0.883i)4-s + (−0.419 + 0.419i)5-s + (−0.308 + 0.0765i)6-s − 0.790i·7-s + (−0.998 − 0.0549i)8-s − 0.899i·9-s + (0.143 + 0.576i)10-s + (0.468 − 0.468i)11-s + (−0.0934 + 0.303i)12-s + (−0.874 − 0.874i)13-s + (−0.677 − 0.407i)14-s + 0.188·15-s + (−0.562 + 0.827i)16-s − 0.467·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.202 - 0.979i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.202 - 0.979i$
Analytic conductor: \(44.7163\)
Root analytic conductor: \(6.68703\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :21/2),\ 0.202 - 0.979i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.4074472512\)
\(L(\frac12)\) \(\approx\) \(0.4074472512\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-746. + 1.24e3i)T \)
good3 \( 1 + (2.29e4 + 2.29e4i)T + 1.04e10iT^{2} \)
5 \( 1 + (9.16e6 - 9.16e6i)T - 4.76e14iT^{2} \)
7 \( 1 + 5.91e8iT - 5.58e17T^{2} \)
11 \( 1 + (-4.03e10 + 4.03e10i)T - 7.40e21iT^{2} \)
13 \( 1 + (4.34e11 + 4.34e11i)T + 2.47e23iT^{2} \)
17 \( 1 + 3.88e12T + 6.90e25T^{2} \)
19 \( 1 + (-9.73e12 - 9.73e12i)T + 7.14e26iT^{2} \)
23 \( 1 - 8.87e13iT - 3.94e28T^{2} \)
29 \( 1 + (6.29e12 + 6.29e12i)T + 5.13e30iT^{2} \)
31 \( 1 + 2.43e15T + 2.08e31T^{2} \)
37 \( 1 + (-2.75e16 + 2.75e16i)T - 8.55e32iT^{2} \)
41 \( 1 + 1.75e16iT - 7.38e33T^{2} \)
43 \( 1 + (9.51e16 - 9.51e16i)T - 2.00e34iT^{2} \)
47 \( 1 - 4.70e17T + 1.30e35T^{2} \)
53 \( 1 + (1.55e18 - 1.55e18i)T - 1.62e36iT^{2} \)
59 \( 1 + (3.87e18 - 3.87e18i)T - 1.54e37iT^{2} \)
61 \( 1 + (-2.76e18 - 2.76e18i)T + 3.10e37iT^{2} \)
67 \( 1 + (-1.01e19 - 1.01e19i)T + 2.22e38iT^{2} \)
71 \( 1 + 3.67e19iT - 7.52e38T^{2} \)
73 \( 1 - 1.88e19iT - 1.34e39T^{2} \)
79 \( 1 + 2.60e19T + 7.08e39T^{2} \)
83 \( 1 + (1.29e20 + 1.29e20i)T + 1.99e40iT^{2} \)
89 \( 1 + 4.10e20iT - 8.65e40T^{2} \)
97 \( 1 + 9.27e20T + 5.27e41T^{2} \)
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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

−12.92340893249407529997788724783, −11.77158726780217121305490463623, −10.70836592739396481849246144430, −9.360688350888656628689682759835, −7.30198415355530784509790988527, −5.80342523227157852159158238906, −4.08178384396871981449707208948, −3.03913923172673389614219281709, −1.17515291152921056598203652935, −0.10512620122993602718028603233, 2.35385097400686212300387563433, 4.30870537915444334103642131743, 5.15915285760729128047343216095, 6.74809502216967002959298838863, 8.150342881414567949879221837650, 9.416637993075844266021323920351, 11.59008285976795841583607111193, 12.58740568513079685613490333911, 14.05662203013875628870083560253, 15.27995024530297527413047346844

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