↑

Properties

Label 2-363-33.8-c1-0-3
Degree $2$
Conductor $363$
Sign $-0.975 - 0.221i$
Analytic cond. $2.89856$
Root an. cond. $1.70251$
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  + (0.535 + 1.64i)3-s + (−0.618 + 1.90i)4-s + (−4.52 − 1.47i)7-s + (−2.42 + 1.76i)9-s − 3.46·12-s + (2.04 + 2.81i)13-s + (−3.23 − 2.35i)16-s + (−1.21 + 0.394i)19-s − 8.24i·21-s + (1.54 + 4.75i)25-s + (−4.20 − 3.05i)27-s + (5.59 − 7.70i)28-s + (1.40 − 1.01i)31-s + (−1.85 − 5.70i)36-s + (−1.60 + 4.94i)37-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)3-s + (−0.309 + 0.951i)4-s + (−1.71 − 0.555i)7-s + (−0.809 + 0.587i)9-s − 0.999·12-s + (0.568 + 0.781i)13-s + (−0.809 − 0.587i)16-s + (−0.278 + 0.0904i)19-s − 1.79i·21-s + (0.309 + 0.951i)25-s + (−0.809 − 0.587i)27-s + (1.05 − 1.45i)28-s + (0.251 − 0.182i)31-s + (−0.309 − 0.951i)36-s + (−0.263 + 0.812i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.975 - 0.221i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ -0.975 - 0.221i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0877151 + 0.783104i\)
\(L(\frac12)\) \(\approx\) \(0.0877151 + 0.783104i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.535 - 1.64i)T \)
11 \( 1 \)
good2 \( 1 + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (4.52 + 1.47i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-2.04 - 2.81i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.21 - 0.394i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.40 + 1.01i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.60 - 4.94i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 13.0iT - 43T^{2} \)
47 \( 1 + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (6.89 - 9.48i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-10.2 - 3.33i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (10.4 + 14.3i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (-4.04 + 2.93i)T + (29.9 - 92.2i)T^{2} \)
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

−11.80516023894239606465380109614, −10.84302608528539761625396006086, −9.771406302465828737458471444824, −9.251195974446050615299636206477, −8.346270186865144148123615935632, −7.15934433666398231658153547678, −6.16251088497008102506803375690, −4.56483047304741057967485392035, −3.67545514188785564664620907186, −2.94878185830129589599033342527, 0.50162964543582439535191902525, 2.35376620134993085794527888312, 3.58749437727563642860022893433, 5.47248883365231842913734885419, 6.23583581743779787105424108516, 6.90282705924735041409082443318, 8.384422870377911790661622849866, 9.121357474214345156458932998477, 9.982363332540805982806987428379, 10.89079770534760102864647897127

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