Properties

Label 2-183-183.182-c0-0-1
Degree $2$
Conductor $183$
Sign $1$
Analytic cond. $0.0913288$
Root an. cond. $0.302206$
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 − 2·13-s + 16-s − 2·19-s + 25-s + 27-s − 36-s − 2·39-s + 48-s + 49-s + 2·52-s − 2·57-s + 61-s − 64-s + 2·73-s + 75-s + 2·76-s + 81-s − 2·97-s − 100-s − 2·103-s − 108-s + 2·109-s − 2·117-s + ⋯
L(s)  = 1  + 3-s − 4-s + 9-s − 12-s − 2·13-s + 16-s − 2·19-s + 25-s + 27-s − 36-s − 2·39-s + 48-s + 49-s + 2·52-s − 2·57-s + 61-s − 64-s + 2·73-s + 75-s + 2·76-s + 81-s − 2·97-s − 100-s − 2·103-s − 108-s + 2·109-s − 2·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(183\)    =    \(3 \cdot 61\)
Sign: $1$
Analytic conductor: \(0.0913288\)
Root analytic conductor: \(0.302206\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{183} (182, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 183,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7276643588\)
\(L(\frac12)\) \(\approx\) \(0.7276643588\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−12.83087761726923673886646212862, −12.39195193396268868922684039319, −10.54369659007926209394245278328, −9.717903375476500356342712797967, −8.859482284770096334717823687742, −8.003018958203834663696043586735, −6.86752148016159565343191931229, −5.01302609419072710833841359669, −4.08407150800302480167473241350, −2.48484983306151831058638012197, 2.48484983306151831058638012197, 4.08407150800302480167473241350, 5.01302609419072710833841359669, 6.86752148016159565343191931229, 8.003018958203834663696043586735, 8.859482284770096334717823687742, 9.717903375476500356342712797967, 10.54369659007926209394245278328, 12.39195193396268868922684039319, 12.83087761726923673886646212862

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