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Properties

Label 2-2e8-32.5-c3-0-18
Degree $2$
Conductor $256$
Sign $-0.892 + 0.450i$
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  + (−3.21 − 7.75i)3-s + (13.6 + 5.65i)5-s + (−9.07 − 9.07i)7-s + (−30.7 + 30.7i)9-s + (16.7 − 40.3i)11-s + (38.4 − 15.9i)13-s − 123. i·15-s − 92.5i·17-s + (−16.6 + 6.90i)19-s + (−41.2 + 99.5i)21-s + (−95.6 + 95.6i)23-s + (65.9 + 65.9i)25-s + (127. + 52.8i)27-s + (−19.3 − 46.7i)29-s + 38.1·31-s + ⋯
L(s)  = 1  + (−0.618 − 1.49i)3-s + (1.22 + 0.505i)5-s + (−0.490 − 0.490i)7-s + (−1.13 + 1.13i)9-s + (0.458 − 1.10i)11-s + (0.819 − 0.339i)13-s − 2.13i·15-s − 1.31i·17-s + (−0.201 + 0.0834i)19-s + (−0.428 + 1.03i)21-s + (−0.866 + 0.866i)23-s + (0.527 + 0.527i)25-s + (0.909 + 0.376i)27-s + (−0.123 − 0.299i)29-s + 0.221·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.450i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.892 + 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.892 + 0.450i$
Analytic conductor: \(15.1044\)
Root analytic conductor: \(3.88644\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :3/2),\ -0.892 + 0.450i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.401677605\)
\(L(\frac12)\) \(\approx\) \(1.401677605\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.21 + 7.75i)T + (-19.0 + 19.0i)T^{2} \)
5 \( 1 + (-13.6 - 5.65i)T + (88.3 + 88.3i)T^{2} \)
7 \( 1 + (9.07 + 9.07i)T + 343iT^{2} \)
11 \( 1 + (-16.7 + 40.3i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + (-38.4 + 15.9i)T + (1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + 92.5iT - 4.91e3T^{2} \)
19 \( 1 + (16.6 - 6.90i)T + (4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (95.6 - 95.6i)T - 1.21e4iT^{2} \)
29 \( 1 + (19.3 + 46.7i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 - 38.1T + 2.97e4T^{2} \)
37 \( 1 + (227. + 94.2i)T + (3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (279. - 279. i)T - 6.89e4iT^{2} \)
43 \( 1 + (-112. + 270. i)T + (-5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 - 321. iT - 1.03e5T^{2} \)
53 \( 1 + (-52.5 + 126. i)T + (-1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (332. + 137. i)T + (1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (33.6 + 81.3i)T + (-1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (-108. - 262. i)T + (-2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (272. + 272. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-372. + 372. i)T - 3.89e5iT^{2} \)
79 \( 1 - 244. iT - 4.93e5T^{2} \)
83 \( 1 + (-1.23e3 + 509. i)T + (4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-216. - 216. i)T + 7.04e5iT^{2} \)
97 \( 1 - 779.T + 9.12e5T^{2} \)
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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

−11.30266974327067290318826221203, −10.37256610010273383715807969378, −9.274691543552700509545923855692, −7.979438469128748676753343291034, −6.86936439633384305384236670612, −6.23190825257447716599279186287, −5.53917712839168126008976073397, −3.28120396048221268392119995404, −1.83953560229794506852363352045, −0.58995909564153030619158996144, 1.86506147804422631587977400523, 3.77732964873111312042029706470, 4.77372407785527719726739868041, 5.81530310383701601655334721365, 6.47208871977916399670127771141, 8.630416859912061989121743073839, 9.310926709084004848869071706850, 10.10139726992071882343128693651, 10.64275600323814486213593215903, 11.97475870920258202964387936868

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