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Properties

Label 2-363-11.9-c1-0-0
Degree $2$
Conductor $363$
Sign $-0.998 - 0.0475i$
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 + 1.53i)2-s + (−0.809 + 0.587i)3-s + (−0.5 − 0.363i)4-s + (−0.118 − 0.363i)5-s + (−0.5 − 1.53i)6-s + (2.42 + 1.76i)7-s + (−1.80 + 1.31i)8-s + (0.309 − 0.951i)9-s + 0.618·10-s + 0.618·12-s + (−1.92 + 5.93i)13-s + (−3.92 + 2.85i)14-s + (0.309 + 0.224i)15-s + (−1.50 − 4.61i)16-s + (0.190 + 0.587i)17-s + (1.30 + 0.951i)18-s + ⋯
L(s)  = 1  + (−0.353 + 1.08i)2-s + (−0.467 + 0.339i)3-s + (−0.250 − 0.181i)4-s + (−0.0527 − 0.162i)5-s + (−0.204 − 0.628i)6-s + (0.917 + 0.666i)7-s + (−0.639 + 0.464i)8-s + (0.103 − 0.317i)9-s + 0.195·10-s + 0.178·12-s + (−0.534 + 1.64i)13-s + (−1.04 + 0.762i)14-s + (0.0797 + 0.0579i)15-s + (−0.375 − 1.15i)16-s + (0.0463 + 0.142i)17-s + (0.308 + 0.224i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.998 - 0.0475i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ -0.998 - 0.0475i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0212164 + 0.892257i\)
\(L(\frac12)\) \(\approx\) \(0.0212164 + 0.892257i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (0.5 - 1.53i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (0.118 + 0.363i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-2.42 - 1.76i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.92 - 5.93i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.190 - 0.587i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.690 - 0.502i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 5.47T + 23T^{2} \)
29 \( 1 + (3.61 + 2.62i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.19 - 3.66i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.42 - 2.48i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.80 + 3.49i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.76T + 43T^{2} \)
47 \( 1 + (-0.5 + 0.363i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.28 - 7.02i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.30 - 3.13i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.354 + 1.08i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 10.5T + 67T^{2} \)
71 \( 1 + (-4.5 - 13.8i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1 - 0.726i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.163 - 0.502i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.92 - 12.0i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 9.47T + 89T^{2} \)
97 \( 1 + (-4.64 + 14.2i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.79551248016442137915330270671, −11.13876108681318117091785551703, −9.762190789013834820405709356923, −8.892041217218605712615860736684, −8.157925404452610575249318927603, −7.10175201479451955716167013271, −6.19671659381611301634911759590, −5.25170572820194156141639314317, −4.26101368014768790561735425307, −2.19377068025384924919431543777, 0.71691118750311535038042708788, 2.13314915279618859584762450438, 3.50395532119027538868892816777, 4.94678110410953670760237073959, 6.08403851355427649296377772560, 7.38260595263824312404188892283, 8.110084534853654804389010326276, 9.495343138584048892861775613752, 10.41416589307503694930774270563, 10.94631369292450859132161787336

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