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Properties

Label 2-363-3.2-c2-0-53
Degree $2$
Conductor $363$
Sign $-1$
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  − 3.31i·2-s + 3·3-s − 7·4-s − 6.63i·5-s − 9.94i·6-s + 8·7-s + 9.94i·8-s + 9·9-s − 22·10-s − 21·12-s − 4·13-s − 26.5i·14-s − 19.8i·15-s + 5.00·16-s − 13.2i·17-s − 29.8i·18-s + ⋯
L(s)  = 1  − 1.65i·2-s + 3-s − 1.75·4-s − 1.32i·5-s − 1.65i·6-s + 1.14·7-s + 1.24i·8-s + 9-s − 2.20·10-s − 1.75·12-s − 0.307·13-s − 1.89i·14-s − 1.32i·15-s + 0.312·16-s − 0.780i·17-s − 1.65i·18-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.38296i\)
\(L(\frac12)\) \(\approx\) \(2.38296i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
11 \( 1 \)
good2 \( 1 + 3.31iT - 4T^{2} \)
5 \( 1 + 6.63iT - 25T^{2} \)
7 \( 1 - 8T + 49T^{2} \)
13 \( 1 + 4T + 169T^{2} \)
17 \( 1 + 13.2iT - 289T^{2} \)
19 \( 1 - 6T + 361T^{2} \)
23 \( 1 - 6.63iT - 529T^{2} \)
29 \( 1 - 39.7iT - 841T^{2} \)
31 \( 1 + 26T + 961T^{2} \)
37 \( 1 - 30T + 1.36e3T^{2} \)
41 \( 1 + 13.2iT - 1.68e3T^{2} \)
43 \( 1 + 42T + 1.84e3T^{2} \)
47 \( 1 - 86.2iT - 2.20e3T^{2} \)
53 \( 1 + 59.6iT - 2.80e3T^{2} \)
59 \( 1 - 66.3iT - 3.48e3T^{2} \)
61 \( 1 + 12T + 3.72e3T^{2} \)
67 \( 1 - 2T + 4.48e3T^{2} \)
71 \( 1 + 59.6iT - 5.04e3T^{2} \)
73 \( 1 - 74T + 5.32e3T^{2} \)
79 \( 1 - 40T + 6.24e3T^{2} \)
83 \( 1 - 39.7iT - 6.88e3T^{2} \)
89 \( 1 + 119. iT - 7.92e3T^{2} \)
97 \( 1 - 62T + 9.40e3T^{2} \)
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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

−10.85938176637635855243928737379, −9.726769499814129162082651552549, −9.102348159074849988871042884137, −8.421573525065110104475712916289, −7.42646468162204052913179520957, −5.05633495519541799269384137125, −4.52576966695752003617016637773, −3.28550989201391704616514474527, −1.95828073273500628633196463102, −1.08156298363179726751424690256, 2.22922139084358807648094135907, 3.78399226838600941140469186207, 4.92557891552064406219353494769, 6.22539578064342916048976025464, 7.10510599803147858989487005421, 7.83190707614531253158941923857, 8.390712779255600836541277311472, 9.525196202594489687419529564589, 10.51396447632304208356989650948, 11.57004167915732868983528299016

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