Properties

Label 2-2e8-32.21-c3-0-16
Degree $2$
Conductor $256$
Sign $0.905 + 0.423i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (9.18 − 3.80i)3-s + (−1.04 + 2.51i)5-s + (16.1 + 16.1i)7-s + (50.8 − 50.8i)9-s + (−3.72 − 1.54i)11-s + (−9.23 − 22.3i)13-s + 27.0i·15-s + 4.95i·17-s + (26.0 + 62.8i)19-s + (209. + 86.6i)21-s + (82.8 − 82.8i)23-s + (83.1 + 83.1i)25-s + (170. − 412. i)27-s + (−150. + 62.3i)29-s − 141.·31-s + ⋯
L(s)  = 1  + (1.76 − 0.732i)3-s + (−0.0932 + 0.225i)5-s + (0.869 + 0.869i)7-s + (1.88 − 1.88i)9-s + (−0.102 − 0.0422i)11-s + (−0.197 − 0.475i)13-s + 0.466i·15-s + 0.0706i·17-s + (0.314 + 0.758i)19-s + (2.17 + 0.900i)21-s + (0.750 − 0.750i)23-s + (0.665 + 0.665i)25-s + (1.21 − 2.93i)27-s + (−0.963 + 0.399i)29-s − 0.820·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.423i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.905 + 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.905 + 0.423i$
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: \((2,\ 256,\ (\ :3/2),\ 0.905 + 0.423i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.557411319\)
\(L(\frac12)\) \(\approx\) \(3.557411319\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9.18 + 3.80i)T + (19.0 - 19.0i)T^{2} \)
5 \( 1 + (1.04 - 2.51i)T + (-88.3 - 88.3i)T^{2} \)
7 \( 1 + (-16.1 - 16.1i)T + 343iT^{2} \)
11 \( 1 + (3.72 + 1.54i)T + (941. + 941. i)T^{2} \)
13 \( 1 + (9.23 + 22.3i)T + (-1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 - 4.95iT - 4.91e3T^{2} \)
19 \( 1 + (-26.0 - 62.8i)T + (-4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (-82.8 + 82.8i)T - 1.21e4iT^{2} \)
29 \( 1 + (150. - 62.3i)T + (1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + 141.T + 2.97e4T^{2} \)
37 \( 1 + (-1.05 + 2.55i)T + (-3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (-8.70 + 8.70i)T - 6.89e4iT^{2} \)
43 \( 1 + (290. + 120. i)T + (5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 + 450. iT - 1.03e5T^{2} \)
53 \( 1 + (-114. - 47.4i)T + (1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (124. - 300. i)T + (-1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (-223. + 92.7i)T + (1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (204. - 84.5i)T + (2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (606. + 606. i)T + 3.57e5iT^{2} \)
73 \( 1 + (531. - 531. i)T - 3.89e5iT^{2} \)
79 \( 1 - 1.12e3iT - 4.93e5T^{2} \)
83 \( 1 + (-118. - 286. i)T + (-4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (-191. - 191. i)T + 7.04e5iT^{2} \)
97 \( 1 + 38.4T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.74592591687839074979242354527, −10.40166680006720275250900701799, −9.163035637356822319624094864763, −8.564537713660860943726636916825, −7.74502560373287182253005037093, −6.91453861162672018390413966996, −5.31833484800180699520296675622, −3.65820639996256273669214919216, −2.60112444446307685014414048039, −1.53685338470527778576427134475, 1.62708577163148116779074393310, 3.02777184020063074441812910867, 4.20138300547445542305215242623, 4.92461013815771007792865760008, 7.20367658446932712694197511887, 7.81541308887780291284059533610, 8.828659815662026314836489195196, 9.501245401874381089291919150531, 10.51463955573453195076902553761, 11.38291137024596946193619496093

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