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Properties

Label 2-363-3.2-c0-0-0
Degree $2$
Conductor $363$
Sign $1$
Analytic cond. $0.181160$
Root an. cond. $0.425629$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s + 9-s − 12-s + 16-s + 25-s − 27-s − 2·31-s + 36-s − 2·37-s − 48-s − 49-s + 64-s − 2·67-s − 75-s + 81-s + 2·93-s + 2·97-s + 100-s + 2·103-s − 108-s + 2·111-s + ⋯
L(s)  = 1  − 3-s + 4-s + 9-s − 12-s + 16-s + 25-s − 27-s − 2·31-s + 36-s − 2·37-s − 48-s − 49-s + 64-s − 2·67-s − 75-s + 81-s + 2·93-s + 2·97-s + 100-s + 2·103-s − 108-s + 2·111-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(0.181160\)
Root analytic conductor: \(0.425629\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{363} (122, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7611828470\)
\(L(\frac12)\) \(\approx\) \(0.7611828470\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
11 \( 1 \)
good2 \( ( 1 - T )( 1 + T ) \)
5 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 + T )^{2} \)
37 \( ( 1 + T )^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T^{2} \)
67 \( ( 1 + T )^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )^{2} \)
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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

−11.57558520424381428387532768529, −10.81261298228495930402358055821, −10.22063685851013016991638995432, −8.947217972865001153286982120980, −7.53280603523901739496442922787, −6.85982008085957695991485271594, −5.91774278653581610914587100242, −4.97384368695213372111039758846, −3.44663079068766757074623151080, −1.74031218766246840331580209164, 1.74031218766246840331580209164, 3.44663079068766757074623151080, 4.97384368695213372111039758846, 5.91774278653581610914587100242, 6.85982008085957695991485271594, 7.53280603523901739496442922787, 8.947217972865001153286982120980, 10.22063685851013016991638995432, 10.81261298228495930402358055821, 11.57558520424381428387532768529

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