Properties

Label 2-363-1.1-c1-0-11
Degree $2$
Conductor $363$
Sign $1$
Analytic cond. $2.89856$
Root an. cond. $1.70251$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3-s + 2·4-s + 4·5-s − 2·6-s − 7-s + 9-s + 8·10-s − 2·12-s + 2·13-s − 2·14-s − 4·15-s − 4·16-s − 4·17-s + 2·18-s + 3·19-s + 8·20-s + 21-s + 2·23-s + 11·25-s + 4·26-s − 27-s − 2·28-s − 6·29-s − 8·30-s − 5·31-s − 8·32-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s + 1.78·5-s − 0.816·6-s − 0.377·7-s + 1/3·9-s + 2.52·10-s − 0.577·12-s + 0.554·13-s − 0.534·14-s − 1.03·15-s − 16-s − 0.970·17-s + 0.471·18-s + 0.688·19-s + 1.78·20-s + 0.218·21-s + 0.417·23-s + 11/5·25-s + 0.784·26-s − 0.192·27-s − 0.377·28-s − 1.11·29-s − 1.46·30-s − 0.898·31-s − 1.41·32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.739014316\)
\(L(\frac12)\) \(\approx\) \(2.739014316\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
11 \( 1 \)
good2 \( 1 - p T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−11.54728253906378153115803392514, −10.75806820233508304777352181849, −9.661688191206149098996496726670, −8.973480572648402934204682616447, −6.98676550080062361288693213459, −6.18537700457688272303978752564, −5.59539380126593624738687647200, −4.73167031165690126954598856077, −3.29412467945553736245288804676, −1.94003785572390378144466057337, 1.94003785572390378144466057337, 3.29412467945553736245288804676, 4.73167031165690126954598856077, 5.59539380126593624738687647200, 6.18537700457688272303978752564, 6.98676550080062361288693213459, 8.973480572648402934204682616447, 9.661688191206149098996496726670, 10.75806820233508304777352181849, 11.54728253906378153115803392514

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