Properties

Label 2-363-1.1-c3-0-45
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  + 4.40·2-s + 3·3-s + 11.4·4-s + 18.8·5-s + 13.2·6-s + 18.5·7-s + 15.1·8-s + 9·9-s + 83.3·10-s + 34.2·12-s − 27.8·13-s + 81.9·14-s + 56.6·15-s − 24.7·16-s − 98.0·17-s + 39.6·18-s − 132.·19-s + 216.·20-s + 55.7·21-s − 180.·23-s + 45.4·24-s + 232.·25-s − 122.·26-s + 27·27-s + 212.·28-s + 132.·29-s + 249.·30-s + ⋯
L(s)  = 1  + 1.55·2-s + 0.577·3-s + 1.42·4-s + 1.69·5-s + 0.899·6-s + 1.00·7-s + 0.668·8-s + 0.333·9-s + 2.63·10-s + 0.825·12-s − 0.593·13-s + 1.56·14-s + 0.975·15-s − 0.386·16-s − 1.39·17-s + 0.519·18-s − 1.60·19-s + 2.41·20-s + 0.579·21-s − 1.64·23-s + 0.386·24-s + 1.85·25-s − 0.925·26-s + 0.192·27-s + 1.43·28-s + 0.849·29-s + 1.52·30-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

\(L(2)\) \(\approx\) \(7.011954543\)
\(L(\frac12)\) \(\approx\) \(7.011954543\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.40T + 8T^{2} \)
5 \( 1 - 18.8T + 125T^{2} \)
7 \( 1 - 18.5T + 343T^{2} \)
13 \( 1 + 27.8T + 2.19e3T^{2} \)
17 \( 1 + 98.0T + 4.91e3T^{2} \)
19 \( 1 + 132.T + 6.85e3T^{2} \)
23 \( 1 + 180.T + 1.21e4T^{2} \)
29 \( 1 - 132.T + 2.43e4T^{2} \)
31 \( 1 - 68.7T + 2.97e4T^{2} \)
37 \( 1 - 170.T + 5.06e4T^{2} \)
41 \( 1 - 259.T + 6.89e4T^{2} \)
43 \( 1 + 49.1T + 7.95e4T^{2} \)
47 \( 1 + 158.T + 1.03e5T^{2} \)
53 \( 1 - 471.T + 1.48e5T^{2} \)
59 \( 1 + 30.9T + 2.05e5T^{2} \)
61 \( 1 + 369.T + 2.26e5T^{2} \)
67 \( 1 - 213.T + 3.00e5T^{2} \)
71 \( 1 - 1.12e3T + 3.57e5T^{2} \)
73 \( 1 + 49.4T + 3.89e5T^{2} \)
79 \( 1 - 615.T + 4.93e5T^{2} \)
83 \( 1 - 100.T + 5.71e5T^{2} \)
89 \( 1 + 322.T + 7.04e5T^{2} \)
97 \( 1 - 1.15e3T + 9.12e5T^{2} \)
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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

−11.12759873475602020725687013395, −10.23762627168602140741816486867, −9.184959966663492532366924933917, −8.192441915636960139716284327834, −6.66047822580691217834369004591, −6.04144296615964825300752378107, −4.90131527281488048692421647305, −4.22342993459894077008416425004, −2.41189745023876140237546714382, −2.04851045105112255784558298752, 2.04851045105112255784558298752, 2.41189745023876140237546714382, 4.22342993459894077008416425004, 4.90131527281488048692421647305, 6.04144296615964825300752378107, 6.66047822580691217834369004591, 8.192441915636960139716284327834, 9.184959966663492532366924933917, 10.23762627168602140741816486867, 11.12759873475602020725687013395

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