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Properties

Label 2-363-11.3-c1-0-5
Degree $2$
Conductor $363$
Sign $-0.469 - 0.882i$
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  + (−2.04 + 1.48i)2-s + (0.309 + 0.951i)3-s + (1.35 − 4.15i)4-s + (1.91 + 1.39i)5-s + (−2.04 − 1.48i)6-s + (0.244 − 0.753i)7-s + (1.85 + 5.69i)8-s + (−0.809 + 0.587i)9-s − 5.98·10-s + 4.37·12-s + (3.32 − 2.41i)13-s + (0.618 + 1.90i)14-s + (−0.733 + 2.25i)15-s + (−5.15 − 3.74i)16-s + (4.84 + 3.51i)17-s + (0.780 − 2.40i)18-s + ⋯
L(s)  = 1  + (−1.44 + 1.04i)2-s + (0.178 + 0.549i)3-s + (0.675 − 2.07i)4-s + (0.858 + 0.623i)5-s + (−0.833 − 0.605i)6-s + (0.0925 − 0.284i)7-s + (0.654 + 2.01i)8-s + (−0.269 + 0.195i)9-s − 1.89·10-s + 1.26·12-s + (0.921 − 0.669i)13-s + (0.165 + 0.508i)14-s + (−0.189 + 0.582i)15-s + (−1.28 − 0.936i)16-s + (1.17 + 0.853i)17-s + (0.183 − 0.565i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.469 - 0.882i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ -0.469 - 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.437980 + 0.729276i\)
\(L(\frac12)\) \(\approx\) \(0.437980 + 0.729276i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.04 - 1.48i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-1.91 - 1.39i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.244 + 0.753i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-3.32 + 2.41i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.84 - 3.51i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.31 - 4.04i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 + (0.780 - 2.40i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.96 - 4.33i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.54 + 4.75i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.75 + 5.41i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.63T + 43T^{2} \)
47 \( 1 + (0.387 + 1.19i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (10.6 - 7.70i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.85 - 5.70i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.16 - 1.57i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 16.1T + 67T^{2} \)
71 \( 1 + (0.602 + 0.437i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.29 + 7.06i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.72 - 3.43i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.88 + 5.00i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 6.37T + 89T^{2} \)
97 \( 1 + (-10.1 + 7.34i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.95683419771391435663636923723, −10.42893318630995941080066121068, −9.872093846007714272317329785958, −8.906686817777419418435616499042, −8.087538333458057374740320944868, −7.19768011275679712593282266712, −6.03617857327197711584350561800, −5.51775393015169485694039942224, −3.46180209454291360759417771235, −1.53109661821332191338751160738, 1.06158214149926780171928478094, 2.08925523399157992313574622494, 3.34160399430885161694738281075, 5.26567385593504985614762797369, 6.65852092395168294077099066747, 7.78217539907197348077994727972, 8.637401021965790771923960447367, 9.405053927887644711336318660416, 9.857148390480482331028153267347, 11.30734986906871240098381816008

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