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Properties

Label 2-363-33.29-c1-0-25
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

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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: $\chi_{363} (161, \cdot )$
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 \( 1 + (-4.04 - 2.93i)T + (29.9 + 92.2i)T^{2} \)
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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

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

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