Properties

Label 2-363-1.1-c3-0-12
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  − 3.37·2-s + 3·3-s + 3.37·4-s − 3.48·5-s − 10.1·6-s + 4.74·7-s + 15.6·8-s + 9·9-s + 11.7·10-s + 10.1·12-s + 15.0·13-s − 16·14-s − 10.4·15-s − 79.6·16-s − 73.1·17-s − 30.3·18-s + 78.7·19-s − 11.7·20-s + 14.2·21-s + 112·23-s + 46.8·24-s − 112.·25-s − 50.6·26-s + 27·27-s + 16·28-s − 243.·29-s + 35.2·30-s + ⋯
L(s)  = 1  − 1.19·2-s + 0.577·3-s + 0.421·4-s − 0.312·5-s − 0.688·6-s + 0.256·7-s + 0.689·8-s + 0.333·9-s + 0.372·10-s + 0.243·12-s + 0.320·13-s − 0.305·14-s − 0.180·15-s − 1.24·16-s − 1.04·17-s − 0.397·18-s + 0.950·19-s − 0.131·20-s + 0.147·21-s + 1.01·23-s + 0.398·24-s − 0.902·25-s − 0.382·26-s + 0.192·27-s + 0.107·28-s − 1.55·29-s + 0.214·30-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

\(L(2)\) \(\approx\) \(1.104507774\)
\(L(\frac12)\) \(\approx\) \(1.104507774\)
\(L(\frac{5}{2})\) 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 \( 1 + 3.37T + 8T^{2} \)
5 \( 1 + 3.48T + 125T^{2} \)
7 \( 1 - 4.74T + 343T^{2} \)
13 \( 1 - 15.0T + 2.19e3T^{2} \)
17 \( 1 + 73.1T + 4.91e3T^{2} \)
19 \( 1 - 78.7T + 6.85e3T^{2} \)
23 \( 1 - 112T + 1.21e4T^{2} \)
29 \( 1 + 243.T + 2.43e4T^{2} \)
31 \( 1 - 278.T + 2.97e4T^{2} \)
37 \( 1 - 102.T + 5.06e4T^{2} \)
41 \( 1 - 241.T + 6.89e4T^{2} \)
43 \( 1 - 280.T + 7.95e4T^{2} \)
47 \( 1 + 169.T + 1.03e5T^{2} \)
53 \( 1 + 409.T + 1.48e5T^{2} \)
59 \( 1 - 196T + 2.05e5T^{2} \)
61 \( 1 - 701.T + 2.26e5T^{2} \)
67 \( 1 - 900.T + 3.00e5T^{2} \)
71 \( 1 - 756.T + 3.57e5T^{2} \)
73 \( 1 - 1.01e3T + 3.89e5T^{2} \)
79 \( 1 - 327.T + 4.93e5T^{2} \)
83 \( 1 - 756.T + 5.71e5T^{2} \)
89 \( 1 - 508.T + 7.04e5T^{2} \)
97 \( 1 - 614.T + 9.12e5T^{2} \)
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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

−10.95020522480700587635653227889, −9.691795110041662432003406481650, −9.240365588721621303282605442657, −8.207484895452689674703071830089, −7.67339622796756854528168352334, −6.61382802346307400930528768102, −4.96418730566925242607158430799, −3.79971373478751606103720714141, −2.22316395953988977196829648052, −0.836177341442496644460259406892, 0.836177341442496644460259406892, 2.22316395953988977196829648052, 3.79971373478751606103720714141, 4.96418730566925242607158430799, 6.61382802346307400930528768102, 7.67339622796756854528168352334, 8.207484895452689674703071830089, 9.240365588721621303282605442657, 9.691795110041662432003406481650, 10.95020522480700587635653227889

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