Properties

Label 2-363-33.29-c1-0-26
Degree $2$
Conductor $363$
Sign $-0.834 - 0.550i$
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  + (−1.42 − 0.987i)3-s + (−0.618 − 1.90i)4-s + (−1.94 − 2.68i)5-s + (1.04 + 2.81i)9-s + (−0.999 + 3.31i)12-s + (0.122 + 5.74i)15-s + (−3.23 + 2.35i)16-s + (−3.89 + 5.36i)20-s + 3.31i·23-s + (−1.85 + 5.70i)25-s + (1.28 − 5.03i)27-s + (−4.04 − 2.93i)31-s + (4.69 − 3.73i)36-s + (−2.16 − 6.65i)37-s + (5.5 − 8.29i)45-s + ⋯
L(s)  = 1  + (−0.821 − 0.570i)3-s + (−0.309 − 0.951i)4-s + (−0.871 − 1.19i)5-s + (0.349 + 0.937i)9-s + (−0.288 + 0.957i)12-s + (0.0316 + 1.48i)15-s + (−0.809 + 0.587i)16-s + (−0.871 + 1.19i)20-s + 0.691i·23-s + (−0.370 + 1.14i)25-s + (0.247 − 0.968i)27-s + (−0.726 − 0.527i)31-s + (0.783 − 0.621i)36-s + (−0.355 − 1.09i)37-s + (0.819 − 1.23i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.834 - 0.550i$
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: \((2,\ 363,\ (\ :1/2),\ -0.834 - 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.109332 + 0.364233i\)
\(L(\frac12)\) \(\approx\) \(0.109332 + 0.364233i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.42 + 0.987i)T \)
11 \( 1 \)
good2 \( 1 + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (1.94 + 2.68i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.31iT - 23T^{2} \)
29 \( 1 + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.04 + 2.93i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.16 + 6.65i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (6.30 + 2.04i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.79 + 10.7i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.15 - 1.02i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 13T + 67T^{2} \)
71 \( 1 + (-9.74 - 13.4i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 16.5iT - 89T^{2} \)
97 \( 1 + (13.7 + 9.99i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.12074216982246988653343388130, −10.02099418655666155221116388127, −9.014401715800957300999672970870, −8.059251452314534486958435555519, −7.03202991712728540758644917210, −5.78171856834322200292317071553, −5.07967363260263247125962434011, −4.12774075435553121452672700218, −1.61149509579184896117696705702, −0.29613251775007733020455175108, 3.03494126603669122292874903642, 3.86653699915197537433462811456, 4.88452191871450324022881804082, 6.42377724536884443481786852698, 7.17388767258288578313929433545, 8.142299878411980864809249756887, 9.265756612124177633522904084021, 10.41798627791991166646296310918, 11.08961107361311522477752488474, 11.88152741904504783026259815783

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