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Properties

Label 2-363-33.29-c1-0-21
Degree $2$
Conductor $363$
Sign $-0.296 + 0.954i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.296 + 0.954i$
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: \((2,\ 363,\ (\ :1/2),\ -0.296 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.895248 - 1.21600i\)
\(L(\frac12)\) \(\approx\) \(0.895248 - 1.21600i\)
\(L(\frac{3}{2})\) 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

−11.27150276007734191489570320509, −10.32171667336956696509804435156, −9.174905545469348679519677344259, −8.283600183533756801277972885244, −7.42623431819028917947464287134, −6.43249179693006835738161811300, −5.21926236191849739735509298042, −4.31111038177601963638085398627, −2.19725535120219912918472779875, −1.12975504792853913981250360425, 2.38285113253565836761139818960, 3.61633370371289869138712173516, 4.82779096027344745192336267278, 5.34534580245777251156940994254, 7.37325998329625116505514027630, 8.212950886000274335341493658731, 8.728544717090808643830944603754, 9.764239411931054814776436676162, 10.91303727180934432960102261919, 11.59416036750641634923799495076

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