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Properties

Label 2-363-33.17-c1-0-10
Degree $2$
Conductor $363$
Sign $0.958 - 0.285i$
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  + (−1.37 + 1.04i)3-s + (1.61 − 1.17i)4-s + (3.15 + 1.02i)5-s + (0.804 − 2.89i)9-s + (−1.00 + 3.31i)12-s + (−5.42 + 1.89i)15-s + (1.23 − 3.80i)16-s + (6.30 − 2.04i)20-s + 3.31i·23-s + (4.85 + 3.52i)25-s + (1.91 + 4.82i)27-s + (1.54 + 4.75i)31-s + (−2.09 − 5.62i)36-s + (5.66 − 4.11i)37-s + (5.5 − 8.29i)45-s + ⋯
L(s)  = 1  + (−0.796 + 0.604i)3-s + (0.809 − 0.587i)4-s + (1.41 + 0.458i)5-s + (0.268 − 0.963i)9-s + (−0.288 + 0.957i)12-s + (−1.40 + 0.488i)15-s + (0.309 − 0.951i)16-s + (1.41 − 0.458i)20-s + 0.691i·23-s + (0.970 + 0.705i)25-s + (0.369 + 0.929i)27-s + (0.277 + 0.854i)31-s + (−0.349 − 0.937i)36-s + (0.931 − 0.676i)37-s + (0.819 − 1.23i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.958 - 0.285i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.958 - 0.285i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54794 + 0.225814i\)
\(L(\frac12)\) \(\approx\) \(1.54794 + 0.225814i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.37 - 1.04i)T \)
11 \( 1 \)
good2 \( 1 + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-3.15 - 1.02i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 3.31iT - 23T^{2} \)
29 \( 1 + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.54 - 4.75i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-5.66 + 4.11i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (3.89 - 5.36i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (12.6 - 4.09i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (1.94 + 2.68i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 13T + 67T^{2} \)
71 \( 1 + (15.7 + 5.12i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 16.5iT - 89T^{2} \)
97 \( 1 + (-5.25 - 16.1i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.20358683597487141127149499264, −10.53023092690842667647959971373, −9.883371787736715552771897994116, −9.178974201320575696355426535556, −7.33886531288004195357491934270, −6.29634572026144167766517492478, −5.86366046478853782796788339137, −4.84627490187079519612064120089, −3.03316022938755799293020086027, −1.58409347437661828091341627436, 1.54085199619496552913239883039, 2.63469280665277238941875095205, 4.64080829661820526060310982090, 5.88868723330624600970806003422, 6.38071434755535904419708441923, 7.45902764118522462052447287009, 8.468099072649252879366378224650, 9.718683341273856977636496625294, 10.59177644915915193551169900028, 11.47271870384123106890290463304

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