Properties

Label 2-363-3.2-c2-0-32
Degree $2$
Conductor $363$
Sign $0.833 - 0.552i$
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  + (2.5 − 1.65i)3-s + 4·4-s + 9.94i·5-s + (3.5 − 8.29i)9-s + (10 − 6.63i)12-s + (16.5 + 24.8i)15-s + 16·16-s + 39.7i·20-s + 29.8i·23-s − 74·25-s + (−4.99 − 26.5i)27-s + 37·31-s + (14 − 33.1i)36-s + 25·37-s + (82.4 + 34.8i)45-s + ⋯
L(s)  = 1  + (0.833 − 0.552i)3-s + 4-s + 1.98i·5-s + (0.388 − 0.921i)9-s + (0.833 − 0.552i)12-s + (1.10 + 1.65i)15-s + 16-s + 1.98i·20-s + 1.29i·23-s − 2.95·25-s + (−0.185 − 0.982i)27-s + 1.19·31-s + (0.388 − 0.921i)36-s + 0.675·37-s + (1.83 + 0.773i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.833 - 0.552i$
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: \((2,\ 363,\ (\ :1),\ 0.833 - 0.552i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.65602 + 0.800820i\)
\(L(\frac12)\) \(\approx\) \(2.65602 + 0.800820i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.5 + 1.65i)T \)
11 \( 1 \)
good2 \( 1 - 4T^{2} \)
5 \( 1 - 9.94iT - 25T^{2} \)
7 \( 1 + 49T^{2} \)
13 \( 1 + 169T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 + 361T^{2} \)
23 \( 1 - 29.8iT - 529T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 - 37T + 961T^{2} \)
37 \( 1 - 25T + 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 1.84e3T^{2} \)
47 \( 1 + 79.5iT - 2.20e3T^{2} \)
53 \( 1 + 79.5iT - 2.80e3T^{2} \)
59 \( 1 - 49.7iT - 3.48e3T^{2} \)
61 \( 1 + 3.72e3T^{2} \)
67 \( 1 + 35T + 4.48e3T^{2} \)
71 \( 1 + 49.7iT - 5.04e3T^{2} \)
73 \( 1 + 5.32e3T^{2} \)
79 \( 1 + 6.24e3T^{2} \)
83 \( 1 - 6.88e3T^{2} \)
89 \( 1 + 149. iT - 7.92e3T^{2} \)
97 \( 1 - 95T + 9.40e3T^{2} \)
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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

−11.40705187730613214962147884641, −10.36640137621970247253454740722, −9.715510668862999087730019259875, −8.128504384806102727876891841569, −7.35938965037992557991093614051, −6.75144386681675944432626640445, −5.98238760160443418989304343754, −3.60283019630482592464062043948, −2.88002660931729143634468657522, −1.91432570704106936968798957549, 1.27822809468157391260321744731, 2.61752170669671940248949119837, 4.16676985653622901857812369999, 4.99063087430731484430696632607, 6.20945163361153676901049389246, 7.75687021152315944090968060667, 8.326368031668718769837317867305, 9.221294436478996552058921796469, 10.02480569420035555321838446259, 11.10753404738016403761117123716

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