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Properties

Label 2-363-33.29-c1-0-14
Degree $2$
Conductor $363$
Sign $-0.892 + 0.450i$
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  + (−1.98 − 1.43i)2-s + (0.360 + 1.69i)3-s + (1.23 + 3.80i)4-s + (−1.01 − 1.40i)5-s + (1.72 − 3.87i)6-s + (−1.34 + 0.437i)7-s + (1.51 − 4.65i)8-s + (−2.74 + 1.22i)9-s + 4.24i·10-s + (−6.00 + 3.46i)12-s + (1.66 − 2.28i)13-s + (3.29 + 1.07i)14-s + (2.00 − 2.22i)15-s + (−3.23 + 2.35i)16-s + (−1.98 + 1.43i)17-s + (7.18 + 1.52i)18-s + ⋯
L(s)  = 1  + (−1.40 − 1.01i)2-s + (0.207 + 0.978i)3-s + (0.618 + 1.90i)4-s + (−0.455 − 0.626i)5-s + (0.704 − 1.58i)6-s + (−0.508 + 0.165i)7-s + (0.535 − 1.64i)8-s + (−0.913 + 0.406i)9-s + 1.34i·10-s + (−1.73 + 0.999i)12-s + (0.461 − 0.634i)13-s + (0.880 + 0.286i)14-s + (0.518 − 0.575i)15-s + (−0.809 + 0.587i)16-s + (−0.480 + 0.349i)17-s + (1.69 + 0.360i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.450i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.892 + 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.892 + 0.450i$
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: \((2,\ 363,\ (\ :1/2),\ -0.892 + 0.450i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0604258 - 0.253794i\)
\(L(\frac12)\) \(\approx\) \(0.0604258 - 0.253794i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.360 - 1.69i)T \)
11 \( 1 \)
good2 \( 1 + (1.98 + 1.43i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (1.01 + 1.40i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (1.34 - 0.437i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.66 + 2.28i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.98 - 1.43i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.68 + 0.874i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 8.66iT - 23T^{2} \)
29 \( 1 + (1.51 + 4.65i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.42 + 1.76i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.927 - 2.85i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.02 - 9.31i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 9.89iT - 43T^{2} \)
47 \( 1 + (6.58 + 2.14i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.03 + 2.80i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.64 - 0.535i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.66 + 2.28i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 3T + 67T^{2} \)
71 \( 1 + (1.01 + 1.40i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-5.37 + 1.74i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.81 + 8.00i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.98 + 1.43i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 8.66iT - 89T^{2} \)
97 \( 1 + (10.5 + 7.64i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.73580444069926860049316775455, −10.16397796627599621282997313025, −9.250150586754857283027270148107, −8.497784064365144914661255748286, −8.063360542774503688368695157120, −6.33452471197538819265540872845, −4.66872564452094681544272384201, −3.55664376072644493195747085586, −2.39959532993522863586884896295, −0.26876651758378505073581399468, 1.62177453264694973474489780574, 3.43241575734793193286621180739, 5.65136894023341846479309984499, 6.68166754987237956074092269247, 7.12429215011102354418650302475, 7.926487488371581453553595275451, 8.905644147753605266738862377728, 9.548952439521635999267964947565, 10.84569247277221502045425192539, 11.45422155202106932637352095596

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