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Properties

Label 2-363-33.5-c2-0-8
Degree $2$
Conductor $363$
Sign $0.929 - 0.369i$
Analytic cond. $9.89103$
Root an. cond. $3.14500$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.04 − 2.81i)3-s + (−3.23 + 2.35i)4-s + (−9.46 − 3.07i)5-s + (−6.80 + 5.89i)9-s + (10 + 6.63i)12-s + (1.27 + 29.8i)15-s + (4.94 − 15.2i)16-s + (37.8 − 12.2i)20-s − 29.8i·23-s + (59.8 + 43.4i)25-s + (23.6 + 12.9i)27-s + (11.4 + 35.1i)31-s + (8.16 − 35.0i)36-s + (−20.2 + 14.6i)37-s + (82.4 − 34.8i)45-s + ⋯
L(s)  = 1  + (−0.349 − 0.937i)3-s + (−0.809 + 0.587i)4-s + (−1.89 − 0.614i)5-s + (−0.756 + 0.654i)9-s + (0.833 + 0.552i)12-s + (0.0848 + 1.98i)15-s + (0.309 − 0.951i)16-s + (1.89 − 0.614i)20-s − 1.29i·23-s + (2.39 + 1.73i)25-s + (0.877 + 0.479i)27-s + (0.368 + 1.13i)31-s + (0.226 − 0.973i)36-s + (−0.546 + 0.397i)37-s + (1.83 − 0.773i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.929 - 0.369i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ 0.929 - 0.369i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.444779 + 0.0852581i\)
\(L(\frac12)\) \(\approx\) \(0.444779 + 0.0852581i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.04 + 2.81i)T \)
11 \( 1 \)
good2 \( 1 + (3.23 - 2.35i)T^{2} \)
5 \( 1 + (9.46 + 3.07i)T + (20.2 + 14.6i)T^{2} \)
7 \( 1 + (15.1 - 46.6i)T^{2} \)
13 \( 1 + (-136. + 99.3i)T^{2} \)
17 \( 1 + (233. + 169. i)T^{2} \)
19 \( 1 + (111. + 343. i)T^{2} \)
23 \( 1 + 29.8iT - 529T^{2} \)
29 \( 1 + (-259. + 799. i)T^{2} \)
31 \( 1 + (-11.4 - 35.1i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (20.2 - 14.6i)T + (423. - 1.30e3i)T^{2} \)
41 \( 1 + (-519. - 1.59e3i)T^{2} \)
43 \( 1 + 1.84e3T^{2} \)
47 \( 1 + (-46.7 + 64.3i)T + (-682. - 2.10e3i)T^{2} \)
53 \( 1 + (75.7 - 24.5i)T + (2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (-29.2 - 40.2i)T + (-1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + 35T + 4.48e3T^{2} \)
71 \( 1 + (-47.3 - 15.3i)T + (4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (-5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 - 149. iT - 7.92e3T^{2} \)
97 \( 1 + (-29.3 - 90.3i)T + (-7.61e3 + 5.53e3i)T^{2} \)
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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

−11.62421701862259320540671407598, −10.63747979606140348064080396276, −8.899717636729964592211635528216, −8.355378981680211818390876303089, −7.66285642162017591474997266120, −6.80771826246713364942020320508, −5.12011538578732466170806701201, −4.30478680018051011428142730455, −3.12560631252409283237293311683, −0.77582668421686330427242968912, 0.35867406250259902248902031471, 3.34992038109763052053836456393, 4.10183051100646912883092893103, 4.95895552153213829248659229704, 6.19524312655588192638157023070, 7.52094914549926966787438883959, 8.437372448375046625891757364404, 9.405117649926992390543916531990, 10.34196580830189215747159868362, 11.16978943114141555921528295126

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