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Properties

Label 2-363-11.4-c1-0-1
Degree $2$
Conductor $363$
Sign $0.659 - 0.751i$
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  + (−0.640 − 0.465i)2-s + (0.309 − 0.951i)3-s + (−0.424 − 1.30i)4-s + (−2.72 + 1.98i)5-s + (−0.640 + 0.465i)6-s + (0.780 + 2.40i)7-s + (−0.825 + 2.54i)8-s + (−0.809 − 0.587i)9-s + 2.67·10-s − 1.37·12-s + (4.72 + 3.43i)13-s + (0.618 − 1.90i)14-s + (1.04 + 3.20i)15-s + (−0.507 + 0.368i)16-s + (−2.16 + 1.57i)17-s + (0.244 + 0.753i)18-s + ⋯
L(s)  = 1  + (−0.453 − 0.329i)2-s + (0.178 − 0.549i)3-s + (−0.212 − 0.652i)4-s + (−1.22 + 0.886i)5-s + (−0.261 + 0.190i)6-s + (0.294 + 0.907i)7-s + (−0.291 + 0.898i)8-s + (−0.269 − 0.195i)9-s + 0.844·10-s − 0.396·12-s + (1.31 + 0.952i)13-s + (0.165 − 0.508i)14-s + (0.269 + 0.828i)15-s + (−0.126 + 0.0922i)16-s + (−0.524 + 0.380i)17-s + (0.0577 + 0.177i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.659 - 0.751i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.659 - 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.621236 + 0.281471i\)
\(L(\frac12)\) \(\approx\) \(0.621236 + 0.281471i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (0.640 + 0.465i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (2.72 - 1.98i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.780 - 2.40i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-4.72 - 3.43i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.16 - 1.57i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.290 - 0.893i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 + (0.244 + 0.753i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.31 + 0.956i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.54 - 4.75i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.36 - 10.3i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 6.63T + 43T^{2} \)
47 \( 1 + (3.93 - 12.1i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.33 - 2.41i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.85 + 5.70i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.84 - 3.51i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 1.11T + 67T^{2} \)
71 \( 1 + (-8.69 + 6.31i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.82 - 8.70i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.32 + 2.41i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.52 + 1.10i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 0.627T + 89T^{2} \)
97 \( 1 + (8.48 + 6.16i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.22385886147795236156437589547, −11.09042336375595841971720268031, −9.653240433288279454726329650336, −8.613001088628259718005862859392, −8.127566293261535037359616237281, −6.76255809709868602627194491776, −6.00415064640630737572942878594, −4.46644554339049063542544908055, −3.06390081618628394003676338688, −1.65280666325032177833019229412, 0.56121466692666824421319245914, 3.50166125042012130682780636727, 4.05059824164538791965083274872, 5.14190955941689848332265825502, 6.92616860817022800516721995721, 7.81807446335303163409949972739, 8.494919337026937123032233108265, 9.056251475346930238196586507052, 10.40881338012991056173598471451, 11.22580558067732439712456195562

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