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Properties

Label 2-363-33.26-c2-0-0
Degree $2$
Conductor $363$
Sign $-0.493 - 0.869i$
Analytic cond. $9.89103$
Root an. cond. $3.14500$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.804 − 2.89i)3-s + (1.23 + 3.80i)4-s + (−5.84 − 8.04i)5-s + (−7.70 + 4.65i)9-s + (9.99 − 6.63i)12-s + (−18.5 + 23.3i)15-s + (−12.9 + 9.40i)16-s + (23.3 − 32.1i)20-s + 29.8i·23-s + (−22.8 + 70.3i)25-s + (19.6 + 18.5i)27-s + (−29.9 − 21.7i)31-s + (−27.2 − 23.5i)36-s + (7.72 + 23.7i)37-s + (82.5 + 34.8i)45-s + ⋯
L(s)  = 1  + (−0.268 − 0.963i)3-s + (0.309 + 0.951i)4-s + (−1.16 − 1.60i)5-s + (−0.856 + 0.516i)9-s + (0.833 − 0.552i)12-s + (−1.23 + 1.55i)15-s + (−0.809 + 0.587i)16-s + (1.16 − 1.60i)20-s + 1.29i·23-s + (−0.914 + 2.81i)25-s + (0.727 + 0.686i)27-s + (−0.965 − 0.701i)31-s + (−0.756 − 0.654i)36-s + (0.208 + 0.642i)37-s + (1.83 + 0.773i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.493 - 0.869i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.493 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.493 - 0.869i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ -0.493 - 0.869i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0383069 + 0.0657827i\)
\(L(\frac12)\) \(\approx\) \(0.0383069 + 0.0657827i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.804 + 2.89i)T \)
11 \( 1 \)
good2 \( 1 + (-1.23 - 3.80i)T^{2} \)
5 \( 1 + (5.84 + 8.04i)T + (-7.72 + 23.7i)T^{2} \)
7 \( 1 + (-39.6 + 28.8i)T^{2} \)
13 \( 1 + (52.2 + 160. i)T^{2} \)
17 \( 1 + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (-292. - 212. i)T^{2} \)
23 \( 1 - 29.8iT - 529T^{2} \)
29 \( 1 + (680. - 494. i)T^{2} \)
31 \( 1 + (29.9 + 21.7i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (-7.72 - 23.7i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (1.35e3 + 988. i)T^{2} \)
43 \( 1 + 1.84e3T^{2} \)
47 \( 1 + (75.7 + 24.5i)T + (1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (46.7 - 64.3i)T + (-868. - 2.67e3i)T^{2} \)
59 \( 1 + (47.3 - 15.3i)T + (2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 + 35T + 4.48e3T^{2} \)
71 \( 1 + (-29.2 - 40.2i)T + (-1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (-4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 + 149. iT - 7.92e3T^{2} \)
97 \( 1 + (76.8 + 55.8i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−11.69135117528457414759433093263, −11.22351638517493176600326560558, −9.278653152646524307403420092095, −8.420305891731577273625774025745, −7.79946917017786796071016477194, −7.14152736242700873133784665987, −5.66430497014266425918713994328, −4.51929791049235712820210233850, −3.36600568746237269420866389190, −1.55675684882983732842631525240, 0.03510110416635932729833286861, 2.65389595159377747391350088693, 3.73974588195325459230189200692, 4.86062257783906000068136296546, 6.19742034487462328741483714107, 6.84714197860669595738244544130, 8.032705208859455283477282589367, 9.342726936773031711139246063470, 10.36716406644405769759399950716, 10.82378252539933121678699593791

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