Properties

Label 2-363-1.1-c3-0-17
Degree $2$
Conductor $363$
Sign $1$
Analytic cond. $21.4176$
Root an. cond. $4.62792$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.63·2-s + 3·3-s + 23.7·4-s + 5.58·5-s − 16.8·6-s + 16.3·7-s − 88.5·8-s + 9·9-s − 31.4·10-s + 71.1·12-s + 36.6·13-s − 91.9·14-s + 16.7·15-s + 308.·16-s + 2.22·17-s − 50.6·18-s + 89.8·19-s + 132.·20-s + 48.9·21-s + 133.·23-s − 265.·24-s − 93.8·25-s − 206.·26-s + 27·27-s + 387.·28-s + 79.6·29-s − 94.3·30-s + ⋯
L(s)  = 1  − 1.99·2-s + 0.577·3-s + 2.96·4-s + 0.499·5-s − 1.14·6-s + 0.881·7-s − 3.91·8-s + 0.333·9-s − 0.994·10-s + 1.71·12-s + 0.782·13-s − 1.75·14-s + 0.288·15-s + 4.82·16-s + 0.0317·17-s − 0.663·18-s + 1.08·19-s + 1.48·20-s + 0.508·21-s + 1.21·23-s − 2.25·24-s − 0.750·25-s − 1.55·26-s + 0.192·27-s + 2.61·28-s + 0.509·29-s − 0.573·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(21.4176\)
Root analytic conductor: \(4.62792\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.278110341\)
\(L(\frac12)\) \(\approx\) \(1.278110341\)
\(L(\frac{5}{2})\) 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 \( 1 + 5.63T + 8T^{2} \)
5 \( 1 - 5.58T + 125T^{2} \)
7 \( 1 - 16.3T + 343T^{2} \)
13 \( 1 - 36.6T + 2.19e3T^{2} \)
17 \( 1 - 2.22T + 4.91e3T^{2} \)
19 \( 1 - 89.8T + 6.85e3T^{2} \)
23 \( 1 - 133.T + 1.21e4T^{2} \)
29 \( 1 - 79.6T + 2.43e4T^{2} \)
31 \( 1 + 167.T + 2.97e4T^{2} \)
37 \( 1 - 38.2T + 5.06e4T^{2} \)
41 \( 1 + 189.T + 6.89e4T^{2} \)
43 \( 1 - 114.T + 7.95e4T^{2} \)
47 \( 1 - 162.T + 1.03e5T^{2} \)
53 \( 1 + 211.T + 1.48e5T^{2} \)
59 \( 1 + 478.T + 2.05e5T^{2} \)
61 \( 1 - 134.T + 2.26e5T^{2} \)
67 \( 1 - 809.T + 3.00e5T^{2} \)
71 \( 1 - 190.T + 3.57e5T^{2} \)
73 \( 1 - 745.T + 3.89e5T^{2} \)
79 \( 1 - 56.5T + 4.93e5T^{2} \)
83 \( 1 + 595.T + 5.71e5T^{2} \)
89 \( 1 - 1.43e3T + 7.04e5T^{2} \)
97 \( 1 + 622.T + 9.12e5T^{2} \)
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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

−10.83452301181893289273125853323, −9.778402345443170059267902059154, −9.162655697746436788588841580929, −8.328021968612733661101324431491, −7.63643024399752479568047197688, −6.69791578370613572341247882103, −5.49511072594376716864572374396, −3.25670937142769234473578939061, −1.98624066776787347617820604462, −1.06000521128116758827770847633, 1.06000521128116758827770847633, 1.98624066776787347617820604462, 3.25670937142769234473578939061, 5.49511072594376716864572374396, 6.69791578370613572341247882103, 7.63643024399752479568047197688, 8.328021968612733661101324431491, 9.162655697746436788588841580929, 9.778402345443170059267902059154, 10.83452301181893289273125853323

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