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Properties

Label 2-363-3.2-c2-0-29
Degree $2$
Conductor $363$
Sign $0.462 - 0.886i$
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  + 3.50i·2-s + (−1.38 + 2.65i)3-s − 8.25·4-s − 7.89i·5-s + (−9.31 − 4.85i)6-s + 1.81·7-s − 14.9i·8-s + (−5.14 − 7.38i)9-s + 27.6·10-s + (11.4 − 21.9i)12-s + 0.0709·13-s + 6.35i·14-s + (21.0 + 10.9i)15-s + 19.1·16-s − 3.66i·17-s + (25.8 − 18.0i)18-s + ⋯
L(s)  = 1  + 1.75i·2-s + (−0.462 + 0.886i)3-s − 2.06·4-s − 1.57i·5-s + (−1.55 − 0.809i)6-s + 0.259·7-s − 1.86i·8-s + (−0.571 − 0.820i)9-s + 2.76·10-s + (0.954 − 1.82i)12-s + 0.00545·13-s + 0.453i·14-s + (1.40 + 0.730i)15-s + 1.19·16-s − 0.215i·17-s + (1.43 − 1.00i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.462 - 0.886i$
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: \((2,\ 363,\ (\ :1),\ 0.462 - 0.886i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.916967 + 0.555808i\)
\(L(\frac12)\) \(\approx\) \(0.916967 + 0.555808i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.38 - 2.65i)T \)
11 \( 1 \)
good2 \( 1 - 3.50iT - 4T^{2} \)
5 \( 1 + 7.89iT - 25T^{2} \)
7 \( 1 - 1.81T + 49T^{2} \)
13 \( 1 - 0.0709T + 169T^{2} \)
17 \( 1 + 3.66iT - 289T^{2} \)
19 \( 1 - 26.3T + 361T^{2} \)
23 \( 1 - 6.84iT - 529T^{2} \)
29 \( 1 + 30.2iT - 841T^{2} \)
31 \( 1 - 42.2T + 961T^{2} \)
37 \( 1 - 3.70T + 1.36e3T^{2} \)
41 \( 1 + 71.0iT - 1.68e3T^{2} \)
43 \( 1 + 13.8T + 1.84e3T^{2} \)
47 \( 1 + 27.3iT - 2.20e3T^{2} \)
53 \( 1 - 21.7iT - 2.80e3T^{2} \)
59 \( 1 + 79.8iT - 3.48e3T^{2} \)
61 \( 1 - 79.1T + 3.72e3T^{2} \)
67 \( 1 + 101.T + 4.48e3T^{2} \)
71 \( 1 + 46.4iT - 5.04e3T^{2} \)
73 \( 1 - 115.T + 5.32e3T^{2} \)
79 \( 1 - 62.8T + 6.24e3T^{2} \)
83 \( 1 - 84.5iT - 6.88e3T^{2} \)
89 \( 1 + 45.3iT - 7.92e3T^{2} \)
97 \( 1 + 96.2T + 9.40e3T^{2} \)
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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

−11.52078018050368506637405545449, −9.911072554417012880061185958956, −9.278259939627686505762657359934, −8.502865799819233192251636746668, −7.71199255999871270383365404632, −6.36820132865457660466619475184, −5.32015153243074429639859564518, −4.96981394118316741435649876338, −3.94523857036193307043372926078, −0.60991896029157343789731929145, 1.24666279018163122996734315004, 2.55967602493741996500640447983, 3.29339328798050303893008089535, 4.87461993275927176962925350269, 6.25778494760788021693651869735, 7.26867320917264179831698378304, 8.332604907888228544619234085175, 9.751936147410024143103137348625, 10.45994918885558474218605311595, 11.27427138207720267914672358191

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