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Properties

Label 2-2e8-16.13-c3-0-22
Degree $2$
Conductor $256$
Sign $-0.923 - 0.382i$
Analytic cond. $15.1044$
Root an. cond. $3.88644$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.51 − 2.51i)3-s + (2.55 − 2.55i)5-s − 2.20i·7-s − 14.3i·9-s + (−31.7 + 31.7i)11-s + (−28.5 − 28.5i)13-s − 12.8·15-s + 87.3·17-s + (−33.6 − 33.6i)19-s + (−5.54 + 5.54i)21-s + 60.0i·23-s + 111. i·25-s + (−104. + 104. i)27-s + (−161. − 161. i)29-s − 323.·31-s + ⋯
L(s)  = 1  + (−0.484 − 0.484i)3-s + (0.228 − 0.228i)5-s − 0.118i·7-s − 0.531i·9-s + (−0.870 + 0.870i)11-s + (−0.609 − 0.609i)13-s − 0.221·15-s + 1.24·17-s + (−0.406 − 0.406i)19-s + (−0.0575 + 0.0575i)21-s + 0.544i·23-s + 0.895i·25-s + (−0.741 + 0.741i)27-s + (−1.03 − 1.03i)29-s − 1.87·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.923 - 0.382i$
Analytic conductor: \(15.1044\)
Root analytic conductor: \(3.88644\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :3/2),\ -0.923 - 0.382i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1931908666\)
\(L(\frac12)\) \(\approx\) \(0.1931908666\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (2.51 + 2.51i)T + 27iT^{2} \)
5 \( 1 + (-2.55 + 2.55i)T - 125iT^{2} \)
7 \( 1 + 2.20iT - 343T^{2} \)
11 \( 1 + (31.7 - 31.7i)T - 1.33e3iT^{2} \)
13 \( 1 + (28.5 + 28.5i)T + 2.19e3iT^{2} \)
17 \( 1 - 87.3T + 4.91e3T^{2} \)
19 \( 1 + (33.6 + 33.6i)T + 6.85e3iT^{2} \)
23 \( 1 - 60.0iT - 1.21e4T^{2} \)
29 \( 1 + (161. + 161. i)T + 2.43e4iT^{2} \)
31 \( 1 + 323.T + 2.97e4T^{2} \)
37 \( 1 + (222. - 222. i)T - 5.06e4iT^{2} \)
41 \( 1 - 133. iT - 6.89e4T^{2} \)
43 \( 1 + (82.9 - 82.9i)T - 7.95e4iT^{2} \)
47 \( 1 + 304.T + 1.03e5T^{2} \)
53 \( 1 + (-40.8 + 40.8i)T - 1.48e5iT^{2} \)
59 \( 1 + (318. - 318. i)T - 2.05e5iT^{2} \)
61 \( 1 + (284. + 284. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-242. - 242. i)T + 3.00e5iT^{2} \)
71 \( 1 + 608. iT - 3.57e5T^{2} \)
73 \( 1 + 582. iT - 3.89e5T^{2} \)
79 \( 1 - 1.15e3T + 4.93e5T^{2} \)
83 \( 1 + (216. + 216. i)T + 5.71e5iT^{2} \)
89 \( 1 + 115. iT - 7.04e5T^{2} \)
97 \( 1 + 940.T + 9.12e5T^{2} \)
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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

−11.12261241609894862721331532027, −10.02192909626402719689260749611, −9.301589453648094546179267201401, −7.78313220633816476480337536339, −7.17311485554276625657177616097, −5.81543261300969324693663603232, −5.04140796661906310833691871576, −3.37872500523926382872930568947, −1.70388204583529606345716662362, −0.07650135148255428198993502651, 2.15648937151591493966742571776, 3.66854200045202984825199185026, 5.14635347243701467119206568380, 5.75611344783230291291923888511, 7.18954130980594165638080332799, 8.207353288524347930356177646788, 9.371579566214895719581151022209, 10.50408293957309953622100267072, 10.83608579761440287835631979695, 12.05622733702109813946281807674

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