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Properties

Label 2-363-11.4-c1-0-11
Degree $2$
Conductor $363$
Sign $0.530 + 0.847i$
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  + (0.5 + 0.363i)2-s + (0.309 − 0.951i)3-s + (−0.5 − 1.53i)4-s + (2.11 − 1.53i)5-s + (0.5 − 0.363i)6-s + (0.927 + 2.85i)7-s + (0.690 − 2.12i)8-s + (−0.809 − 0.587i)9-s + 1.61·10-s − 1.61·12-s + (−1.42 − 1.03i)13-s + (−0.572 + 1.76i)14-s + (−0.809 − 2.48i)15-s + (−1.49 + 1.08i)16-s + (−1.30 + 0.951i)17-s + (−0.190 − 0.587i)18-s + ⋯
L(s)  = 1  + (0.353 + 0.256i)2-s + (0.178 − 0.549i)3-s + (−0.250 − 0.769i)4-s + (0.947 − 0.688i)5-s + (0.204 − 0.148i)6-s + (0.350 + 1.07i)7-s + (0.244 − 0.751i)8-s + (−0.269 − 0.195i)9-s + 0.511·10-s − 0.467·12-s + (−0.395 − 0.287i)13-s + (−0.153 + 0.471i)14-s + (−0.208 − 0.642i)15-s + (−0.374 + 0.272i)16-s + (−0.317 + 0.230i)17-s + (−0.0450 − 0.138i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.530 + 0.847i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.530 + 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62158 - 0.898563i\)
\(L(\frac12)\) \(\approx\) \(1.62158 - 0.898563i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (-0.5 - 0.363i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-2.11 + 1.53i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.927 - 2.85i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.42 + 1.03i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.30 - 0.951i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.80 + 5.56i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 3.47T + 23T^{2} \)
29 \( 1 + (-1.38 - 4.25i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.30 + 1.67i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.0729 - 0.224i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.69 - 11.3i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 6.23T + 43T^{2} \)
47 \( 1 + (-0.5 + 1.53i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.78 - 5.65i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.19 - 9.82i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (6.35 - 4.61i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 9.56T + 67T^{2} \)
71 \( 1 + (-4.5 + 3.26i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1 + 3.07i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (7.66 + 5.56i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.572 - 0.416i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 0.527T + 89T^{2} \)
97 \( 1 + (-11.3 - 8.24i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.38395212513297788514707827976, −10.24080367972536100250183533881, −9.075821379581513942487330338635, −8.923024790272086637771250123124, −7.33755965728790733463645999066, −6.20775324840554611490747189686, −5.42880543489262541361114848235, −4.74790922346036442633292028271, −2.60811232411996846543507672799, −1.31746970785559094205319291544, 2.20853427167477797470569249881, 3.46679844276944781304593954070, 4.38550666087727332738719889118, 5.52225266209572139729997633962, 6.93525815413676148858908807859, 7.76885433739258316904409955981, 8.914209605495293270090296173582, 9.937572972753954476326543351681, 10.60690522095951625559704126104, 11.47328398392640637050507012359

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