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Properties

Label 2-363-33.20-c2-0-38
Degree $2$
Conductor $363$
Sign $-0.719 + 0.694i$
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  + (−3.15 − 1.02i)2-s + (−2.42 + 1.76i)3-s + (5.66 + 4.11i)4-s + (6.30 − 2.04i)5-s + (9.46 − 3.07i)6-s + (−6.47 − 4.70i)7-s + (−5.84 − 8.04i)8-s + (2.78 − 8.55i)9-s − 22·10-s − 21.0·12-s + (−1.23 + 3.80i)13-s + (15.5 + 21.4i)14-s + (−11.6 + 16.0i)15-s + (1.54 + 4.75i)16-s + (12.6 − 4.09i)17-s + (−17.5 + 24.1i)18-s + ⋯
L(s)  = 1  + (−1.57 − 0.512i)2-s + (−0.809 + 0.587i)3-s + (1.41 + 1.02i)4-s + (1.26 − 0.409i)5-s + (1.57 − 0.512i)6-s + (−0.924 − 0.671i)7-s + (−0.731 − 1.00i)8-s + (0.309 − 0.951i)9-s − 2.20·10-s − 1.75·12-s + (−0.0950 + 0.292i)13-s + (1.11 + 1.53i)14-s + (−0.779 + 1.07i)15-s + (0.0965 + 0.297i)16-s + (0.742 − 0.241i)17-s + (−0.974 + 1.34i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.719 + 0.694i$
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: \((2,\ 363,\ (\ :1),\ -0.719 + 0.694i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.139670 - 0.345975i\)
\(L(\frac12)\) \(\approx\) \(0.139670 - 0.345975i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.42 - 1.76i)T \)
11 \( 1 \)
good2 \( 1 + (3.15 + 1.02i)T + (3.23 + 2.35i)T^{2} \)
5 \( 1 + (-6.30 + 2.04i)T + (20.2 - 14.6i)T^{2} \)
7 \( 1 + (6.47 + 4.70i)T + (15.1 + 46.6i)T^{2} \)
13 \( 1 + (1.23 - 3.80i)T + (-136. - 99.3i)T^{2} \)
17 \( 1 + (-12.6 + 4.09i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (4.85 - 3.52i)T + (111. - 343. i)T^{2} \)
23 \( 1 - 6.63iT - 529T^{2} \)
29 \( 1 + (-23.3 + 32.1i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (8.03 - 24.7i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (24.2 + 17.6i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (-7.79 - 10.7i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 42T + 1.84e3T^{2} \)
47 \( 1 + (50.6 + 69.7i)T + (-682. + 2.10e3i)T^{2} \)
53 \( 1 + (56.7 + 18.4i)T + (2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (-38.9 + 53.6i)T + (-1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (3.70 + 11.4i)T + (-3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 - 2T + 4.48e3T^{2} \)
71 \( 1 + (-56.7 + 18.4i)T + (4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (59.8 + 43.4i)T + (1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (-12.3 + 38.0i)T + (-5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (37.8 - 12.2i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 + 119. iT - 7.92e3T^{2} \)
97 \( 1 + (-19.1 + 58.9i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−10.38337022717241757495024248947, −9.900447155655385318658103004434, −9.523676586010378801195116532703, −8.477066751873401536856249876519, −7.05716831574332147636202162747, −6.24354288029026827876109352949, −5.04173827969910618898021394081, −3.36422530565556940997342845762, −1.66892633947050941979797349487, −0.33062370104172192175920274758, 1.36965477048823172204803280569, 2.63127881566113622619271851729, 5.37587793179053686057817274970, 6.26736460339532667110849931338, 6.66175794361035265859364653796, 7.79835258921240685641462329483, 8.862778880581841804220576638767, 9.830382909464908826862362489300, 10.25392865377868482023362899292, 11.16502525207825612298593289518

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