Properties

Label 2-864-216.155-c1-0-26
Degree $2$
Conductor $864$
Sign $0.382 + 0.924i$
Analytic cond. $6.89907$
Root an. cond. $2.62660$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.56 − 0.739i)3-s + (1.90 − 2.31i)9-s + (−5.93 − 1.04i)11-s + (6.38 − 3.68i)17-s + (4.35 − 7.53i)19-s + (4.69 − 1.71i)25-s + (1.27 − 5.03i)27-s + (−10.0 + 2.74i)33-s + (−1.34 + 3.70i)41-s + (−1.12 + 6.39i)43-s + (1.21 + 6.89i)49-s + (7.27 − 10.4i)51-s + (1.24 − 15.0i)57-s + (−8.23 + 1.45i)59-s + (4.95 + 1.80i)67-s + ⋯
L(s)  = 1  + (0.904 − 0.426i)3-s + (0.635 − 0.771i)9-s + (−1.78 − 0.315i)11-s + (1.54 − 0.893i)17-s + (0.998 − 1.72i)19-s + (0.939 − 0.342i)25-s + (0.245 − 0.969i)27-s + (−1.75 + 0.478i)33-s + (−0.210 + 0.578i)41-s + (−0.171 + 0.975i)43-s + (0.173 + 0.984i)49-s + (1.01 − 1.46i)51-s + (0.164 − 1.98i)57-s + (−1.07 + 0.189i)59-s + (0.605 + 0.220i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.382 + 0.924i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.382 + 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66344 - 1.11232i\)
\(L(\frac12)\) \(\approx\) \(1.66344 - 1.11232i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.56 + 0.739i)T \)
good5 \( 1 + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-1.21 - 6.89i)T^{2} \)
11 \( 1 + (5.93 + 1.04i)T + (10.3 + 3.76i)T^{2} \)
13 \( 1 + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-6.38 + 3.68i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.35 + 7.53i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-5.38 + 30.5i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.34 - 3.70i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.12 - 6.39i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (8.23 - 1.45i)T + (55.4 - 20.1i)T^{2} \)
61 \( 1 + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (-4.95 - 1.80i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.96 - 3.39i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-4.84 - 13.3i)T + (-63.5 + 53.3i)T^{2} \)
89 \( 1 + (15.9 + 9.20i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.564 + 3.19i)T + (-91.1 - 33.1i)T^{2} \)
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

−9.837944528616495694921023293213, −9.184376525216262687497141514360, −8.137430714111689843421439961581, −7.61691906949215539911380924621, −6.84023088693411483782798691387, −5.50808825114994452766233562345, −4.71373840252333643842799520100, −3.02930349737491704975599698580, −2.76942870918805727398866714684, −0.926125754696394381975231264519, 1.71189659887162946773567358716, 3.00199360021158503187462399633, 3.73414578689880761929411617792, 5.08020416396331300833340609515, 5.67564857284509378896303396648, 7.28616153083850510988762681145, 7.897546091343641987184321233907, 8.459007228416718413003560024841, 9.663011832848004186994751171417, 10.24905442853845839910576045872

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