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Properties

Label 2-363-33.20-c2-0-42
Degree $2$
Conductor $363$
Sign $0.0206 + 0.999i$
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  + (−2.99 + 0.127i)3-s + (−3.23 − 2.35i)4-s + (9.46 − 3.07i)5-s + (8.96 − 0.766i)9-s + (10 + 6.63i)12-s + (−27.9 + 10.4i)15-s + (4.94 + 15.2i)16-s + (−37.8 − 12.2i)20-s − 29.8i·23-s + (59.8 − 43.4i)25-s + (−26.7 + 3.44i)27-s + (11.4 − 35.1i)31-s + (−30.8 − 18.6i)36-s + (−20.2 − 14.6i)37-s + (82.4 − 34.8i)45-s + ⋯
L(s)  = 1  + (−0.999 + 0.0426i)3-s + (−0.809 − 0.587i)4-s + (1.89 − 0.614i)5-s + (0.996 − 0.0851i)9-s + (0.833 + 0.552i)12-s + (−1.86 + 0.695i)15-s + (0.309 + 0.951i)16-s + (−1.89 − 0.614i)20-s − 1.29i·23-s + (2.39 − 1.73i)25-s + (−0.991 + 0.127i)27-s + (0.368 − 1.13i)31-s + (−0.856 − 0.516i)36-s + (−0.546 − 0.397i)37-s + (1.83 − 0.773i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0206 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0206 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.0206 + 0.999i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ 0.0206 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.920473 - 0.901662i\)
\(L(\frac12)\) \(\approx\) \(0.920473 - 0.901662i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.99 - 0.127i)T \)
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

−10.57500912116286141186686904683, −10.14026441911381953999793858805, −9.378072369057674714551476251751, −8.542539766849665301275043649962, −6.68910879623552099374890798759, −5.89703156485568924672050161457, −5.25083676705035413266976836548, −4.40325368742249465968971933908, −1.98663282712181030875322580249, −0.72967833183705970713979538950, 1.51474209413340332609270905183, 3.15682398245965353697259572245, 4.79713485758522555622101550779, 5.59228685583239676191217308905, 6.47549660821507802581284148223, 7.45644834543155895348333639684, 8.957697927215268409103345871995, 9.758755466308655328838001170886, 10.35345599054924049161016589445, 11.37531955961604794134966640667

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