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Properties

Label 2-363-11.5-c1-0-17
Degree $2$
Conductor $363$
Sign $0.569 - 0.821i$
Analytic cond. $2.89856$
Root an. cond. $1.70251$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.690 − 2.12i)2-s + (−0.809 − 0.587i)3-s + (−2.42 + 1.76i)4-s + (0.618 − 1.90i)5-s + (−0.690 + 2.12i)6-s + (−3.61 + 2.62i)7-s + (1.80 + 1.31i)8-s + (0.309 + 0.951i)9-s − 4.47·10-s + 3·12-s + (8.09 + 5.87i)14-s + (−1.61 + 1.17i)15-s + (−0.309 + 0.951i)16-s + (−1.38 + 4.25i)17-s + (1.80 − 1.31i)18-s + (−3.61 − 2.62i)19-s + ⋯
L(s)  = 1  + (−0.488 − 1.50i)2-s + (−0.467 − 0.339i)3-s + (−1.21 + 0.881i)4-s + (0.276 − 0.850i)5-s + (−0.282 + 0.868i)6-s + (−1.36 + 0.993i)7-s + (0.639 + 0.464i)8-s + (0.103 + 0.317i)9-s − 1.41·10-s + 0.866·12-s + (2.16 + 1.57i)14-s + (−0.417 + 0.303i)15-s + (−0.0772 + 0.237i)16-s + (−0.335 + 1.03i)17-s + (0.426 − 0.309i)18-s + (−0.830 − 0.603i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.569 - 0.821i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.569 - 0.821i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (0.690 + 2.12i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-0.618 + 1.90i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (3.61 - 2.62i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.38 - 4.25i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.61 + 2.62i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + (-3.61 + 2.62i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.61 - 1.17i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.61 + 2.62i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 4.47T + 43T^{2} \)
47 \( 1 + (6.47 + 4.70i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.85 - 5.70i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.76 - 8.50i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + (2.47 - 7.60i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (7.23 - 5.25i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (4.14 + 12.7i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.76 + 8.50i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 + (-0.618 - 1.90i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.55474462393294047640874709830, −9.925112454718506083738898296916, −8.944652116474531204865782281404, −8.488713369923183173420331367884, −6.60539953686446730429116473884, −5.76349084039087260304377611668, −4.30780903858763957364065878385, −2.91730361629439287662523984329, −1.74997079322915784401467708903, 0, 3.16544726887110387131289330136, 4.57766661514690438307229868508, 5.95499319061068649806064755413, 6.65347699744298434320375230247, 7.08636131381566549613954010335, 8.316925767634156724823786351147, 9.562941994497872692304665891305, 10.05348452180264706994765430231, 10.89586377851748224568865396089

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