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Properties

Label 2-363-33.17-c1-0-11
Degree $2$
Conductor $363$
Sign $0.293 - 0.955i$
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  + (0.465 + 1.43i)2-s + (1.73 − 0.0585i)3-s + (−0.216 + 0.157i)4-s + (−2.76 − 0.899i)5-s + (0.889 + 2.45i)6-s + (1.66 + 2.28i)7-s + (2.11 + 1.53i)8-s + (2.99 − 0.202i)9-s − 4.38i·10-s + (−0.366 + 0.285i)12-s + (0.852 − 0.277i)13-s + (−2.50 + 3.44i)14-s + (−4.84 − 1.39i)15-s + (−1.37 + 4.24i)16-s + (0.806 − 2.48i)17-s + (1.68 + 4.19i)18-s + ⋯
L(s)  = 1  + (0.329 + 1.01i)2-s + (0.999 − 0.0338i)3-s + (−0.108 + 0.0787i)4-s + (−1.23 − 0.402i)5-s + (0.363 + 1.00i)6-s + (0.628 + 0.864i)7-s + (0.746 + 0.542i)8-s + (0.997 − 0.0676i)9-s − 1.38i·10-s + (−0.105 + 0.0823i)12-s + (0.236 − 0.0768i)13-s + (−0.669 + 0.920i)14-s + (−1.25 − 0.359i)15-s + (−0.344 + 1.06i)16-s + (0.195 − 0.601i)17-s + (0.396 + 0.988i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.293 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.293 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.293 - 0.955i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.293 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71522 + 1.26736i\)
\(L(\frac12)\) \(\approx\) \(1.71522 + 1.26736i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 + 0.0585i)T \)
11 \( 1 \)
good2 \( 1 + (-0.465 - 1.43i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (2.76 + 0.899i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (-1.66 - 2.28i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.852 + 0.277i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.806 + 2.48i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.43 - 1.98i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (6.98 - 5.07i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.15 + 9.69i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.86 + 2.07i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.65 + 2.65i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 4.24iT - 43T^{2} \)
47 \( 1 + (-1.25 + 1.72i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (6.27 - 2.03i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.335 + 0.461i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (9.67 + 3.14i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + (-9.04 - 2.93i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.88 + 2.59i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (6.98 - 2.27i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.95 + 6.00i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 12.4iT - 89T^{2} \)
97 \( 1 + (-1.77 - 5.45i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.70443629145968233422833243472, −10.84024087578547819831964369279, −9.339281834039968237075310044180, −8.456845358881814933589168586240, −7.82431392430435344444220756114, −7.19738187152923104489349547278, −5.77224757651116677470294910396, −4.72132195529674481934762841808, −3.70095923173038249937760424551, −2.01055732822597808912045259506, 1.56326264474683610059526092755, 3.07037358374727738722199685889, 3.87489304126639566681534333562, 4.54514481517483570004821039544, 6.87935495185210465593030103942, 7.63352447056399665119523100232, 8.263237688982885717338264621146, 9.610395564521321998533340472923, 10.73721072356598661505937579665, 11.07707638440668516969543892109

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