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Properties

Label 2-363-3.2-c2-0-13
Degree $2$
Conductor $363$
Sign $0.0421 + 0.999i$
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  + 3.06i·2-s + (0.126 + 2.99i)3-s − 5.36·4-s + 6.63i·5-s + (−9.17 + 0.386i)6-s − 3.06·7-s − 4.17i·8-s + (−8.96 + 0.757i)9-s − 20.3·10-s + (−0.678 − 16.0i)12-s + 16.2·13-s − 9.37i·14-s + (−19.9 + 0.839i)15-s − 8.67·16-s − 0.805i·17-s + (−2.31 − 27.4i)18-s + ⋯
L(s)  = 1  + 1.53i·2-s + (0.0421 + 0.999i)3-s − 1.34·4-s + 1.32i·5-s + (−1.52 + 0.0644i)6-s − 0.437·7-s − 0.522i·8-s + (−0.996 + 0.0842i)9-s − 2.03·10-s + (−0.0565 − 1.34i)12-s + 1.24·13-s − 0.669i·14-s + (−1.32 + 0.0559i)15-s − 0.542·16-s − 0.0473i·17-s + (−0.128 − 1.52i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0421 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0421 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.0421 + 0.999i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ 0.0421 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.958388 - 0.918812i\)
\(L(\frac12)\) \(\approx\) \(0.958388 - 0.918812i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.126 - 2.99i)T \)
11 \( 1 \)
good2 \( 1 - 3.06iT - 4T^{2} \)
5 \( 1 - 6.63iT - 25T^{2} \)
7 \( 1 + 3.06T + 49T^{2} \)
13 \( 1 - 16.2T + 169T^{2} \)
17 \( 1 + 0.805iT - 289T^{2} \)
19 \( 1 - 20.7T + 361T^{2} \)
23 \( 1 - 27.3iT - 529T^{2} \)
29 \( 1 + 3.78iT - 841T^{2} \)
31 \( 1 + 20.7T + 961T^{2} \)
37 \( 1 - 38.5T + 1.36e3T^{2} \)
41 \( 1 + 13.3iT - 1.68e3T^{2} \)
43 \( 1 - 43.4T + 1.84e3T^{2} \)
47 \( 1 - 19.8iT - 2.20e3T^{2} \)
53 \( 1 + 17.6iT - 2.80e3T^{2} \)
59 \( 1 - 43.5iT - 3.48e3T^{2} \)
61 \( 1 - 10.6T + 3.72e3T^{2} \)
67 \( 1 - 72.2T + 4.48e3T^{2} \)
71 \( 1 - 2.56iT - 5.04e3T^{2} \)
73 \( 1 + 45.4T + 5.32e3T^{2} \)
79 \( 1 + 98.2T + 6.24e3T^{2} \)
83 \( 1 + 31.9iT - 6.88e3T^{2} \)
89 \( 1 - 18.5iT - 7.92e3T^{2} \)
97 \( 1 + 63.3T + 9.40e3T^{2} \)
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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

−11.48221318766124299562623694759, −10.95172948469297567279016143518, −9.823325038500020163834288364459, −9.089226614396428341597564459055, −7.973480265500569469382244343397, −7.09407996193678816070244425735, −6.11707669580356856536814001674, −5.48578316285756986756716344644, −4.00184727532026384677513303579, −3.00656895534658703299893137475, 0.64371247556165195330443826624, 1.49043015313875821702785336318, 2.88244643573784140699139865527, 4.08942042470303363911223621763, 5.41899444056470618444310945086, 6.60613154185277710964276638375, 8.050256299421765614802281551505, 8.881964725989756782639959345269, 9.548430650796845532100220375306, 10.82481746781368033615124886383

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