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Properties

Label 2-2e8-32.21-c3-0-4
Degree $2$
Conductor $256$
Sign $-0.969 + 0.245i$
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  + (−5.56 + 2.30i)3-s + (−6.28 + 15.1i)5-s + (16.6 + 16.6i)7-s + (6.60 − 6.60i)9-s + (3.11 + 1.28i)11-s + (28.8 + 69.7i)13-s − 98.9i·15-s + 66.2i·17-s + (−12.5 − 30.2i)19-s + (−131. − 54.3i)21-s + (63.7 − 63.7i)23-s + (−102. − 102. i)25-s + (40.7 − 98.3i)27-s + (−190. + 79.0i)29-s − 123.·31-s + ⋯
L(s)  = 1  + (−1.07 + 0.443i)3-s + (−0.562 + 1.35i)5-s + (0.899 + 0.899i)7-s + (0.244 − 0.244i)9-s + (0.0853 + 0.0353i)11-s + (0.616 + 1.48i)13-s − 1.70i·15-s + 0.944i·17-s + (−0.151 − 0.365i)19-s + (−1.36 − 0.564i)21-s + (0.577 − 0.577i)23-s + (−0.818 − 0.818i)25-s + (0.290 − 0.701i)27-s + (−1.22 + 0.506i)29-s − 0.717·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.969 + 0.245i$
Analytic conductor: \(15.1044\)
Root analytic conductor: \(3.88644\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :3/2),\ -0.969 + 0.245i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8715297079\)
\(L(\frac12)\) \(\approx\) \(0.8715297079\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.56 - 2.30i)T + (19.0 - 19.0i)T^{2} \)
5 \( 1 + (6.28 - 15.1i)T + (-88.3 - 88.3i)T^{2} \)
7 \( 1 + (-16.6 - 16.6i)T + 343iT^{2} \)
11 \( 1 + (-3.11 - 1.28i)T + (941. + 941. i)T^{2} \)
13 \( 1 + (-28.8 - 69.7i)T + (-1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 - 66.2iT - 4.91e3T^{2} \)
19 \( 1 + (12.5 + 30.2i)T + (-4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (-63.7 + 63.7i)T - 1.21e4iT^{2} \)
29 \( 1 + (190. - 79.0i)T + (1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + 123.T + 2.97e4T^{2} \)
37 \( 1 + (-46.0 + 111. i)T + (-3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (100. - 100. i)T - 6.89e4iT^{2} \)
43 \( 1 + (-27.5 - 11.4i)T + (5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 - 394. iT - 1.03e5T^{2} \)
53 \( 1 + (135. + 56.0i)T + (1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (-297. + 717. i)T + (-1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (-548. + 227. i)T + (1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (-163. + 67.6i)T + (2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (194. + 194. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-547. + 547. i)T - 3.89e5iT^{2} \)
79 \( 1 - 715. iT - 4.93e5T^{2} \)
83 \( 1 + (-54.6 - 131. i)T + (-4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (-220. - 220. i)T + 7.04e5iT^{2} \)
97 \( 1 - 1.36e3T + 9.12e5T^{2} \)
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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

−11.55017210431178412943747321610, −11.21754746161415629258161287799, −10.69522011938760159874891028447, −9.268044124586708225942267792221, −8.173591694606270522923326081283, −6.87854071779300522163753512724, −6.09114255507120917136518470300, −4.92883265495548463155138731766, −3.75862305700424748732877027423, −2.07246020185649898674349745027, 0.45860567927875320133678519465, 1.20752033318166911909323180605, 3.79885595833178935107651078760, 5.05034281053363547547149019258, 5.62944425681759530160327607476, 7.20881078496503577670167173464, 7.980835769995833142316492035392, 8.953575776189050122937886866996, 10.39144211162648739499233691983, 11.33097500356662575760479898424

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