Properties

Label 2-864-32.13-c1-0-12
Degree $2$
Conductor $864$
Sign $-0.955 - 0.294i$
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  + (−0.139 + 1.40i)2-s + (−1.96 − 0.393i)4-s + (−0.345 + 0.143i)5-s + (0.299 − 0.299i)7-s + (0.827 − 2.70i)8-s + (−0.153 − 0.506i)10-s + (1.61 + 3.90i)11-s + (−1.58 − 0.655i)13-s + (0.379 + 0.462i)14-s + (3.69 + 1.54i)16-s + 1.95i·17-s + (0.0281 + 0.0116i)19-s + (0.734 − 0.144i)20-s + (−5.71 + 1.72i)22-s + (2.80 + 2.80i)23-s + ⋯
L(s)  = 1  + (−0.0988 + 0.995i)2-s + (−0.980 − 0.196i)4-s + (−0.154 + 0.0640i)5-s + (0.113 − 0.113i)7-s + (0.292 − 0.956i)8-s + (−0.0484 − 0.160i)10-s + (0.487 + 1.17i)11-s + (−0.438 − 0.181i)13-s + (0.101 + 0.123i)14-s + (0.922 + 0.385i)16-s + 0.473i·17-s + (0.00646 + 0.00267i)19-s + (0.164 − 0.0323i)20-s + (−1.21 + 0.368i)22-s + (0.583 + 0.583i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.955 - 0.294i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ -0.955 - 0.294i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.140219 + 0.932304i\)
\(L(\frac12)\) \(\approx\) \(0.140219 + 0.932304i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.139 - 1.40i)T \)
3 \( 1 \)
good5 \( 1 + (0.345 - 0.143i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-0.299 + 0.299i)T - 7iT^{2} \)
11 \( 1 + (-1.61 - 3.90i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (1.58 + 0.655i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 1.95iT - 17T^{2} \)
19 \( 1 + (-0.0281 - 0.0116i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-2.80 - 2.80i)T + 23iT^{2} \)
29 \( 1 + (1.57 - 3.79i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 7.50T + 31T^{2} \)
37 \( 1 + (2.16 - 0.896i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-6.80 - 6.80i)T + 41iT^{2} \)
43 \( 1 + (2.40 + 5.81i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 2.05iT - 47T^{2} \)
53 \( 1 + (1.75 + 4.23i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (11.3 - 4.71i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (2.70 - 6.53i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-0.0765 + 0.184i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-4.37 + 4.37i)T - 71iT^{2} \)
73 \( 1 + (-2.81 - 2.81i)T + 73iT^{2} \)
79 \( 1 + 6.03iT - 79T^{2} \)
83 \( 1 + (-2.55 - 1.06i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-2.16 + 2.16i)T - 89iT^{2} \)
97 \( 1 - 1.88T + 97T^{2} \)
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

−10.31181582648543013364764203985, −9.453605464662251072831040292660, −8.926930902427506010483574080696, −7.60366514795749275045114408486, −7.38020827340732100806493745813, −6.32253936971383109229725872995, −5.33139250393865979025537976496, −4.48395652981116417620819243017, −3.49198132938476511415486555380, −1.60406285415176396962659783815, 0.48953856300407255826823033719, 2.04713323111334413074262931089, 3.21138589420424528337532393627, 4.13125833460551879263278405090, 5.15861754468711135509090094493, 6.14499106753957819742570861695, 7.44758716330904994527045622403, 8.353513353291695250369522521388, 9.093749119074552324188659016257, 9.756633531980425592426395875393

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