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Properties

Label 2-184-184.45-c0-0-0
Degree $2$
Conductor $184$
Sign $1$
Analytic cond. $0.0918279$
Root an. cond. $0.303031$
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  − 2-s + 4-s − 8-s + 9-s + 16-s − 18-s − 23-s − 25-s − 2·31-s − 32-s + 36-s − 2·41-s + 46-s + 2·47-s + 49-s + 50-s + 2·62-s + 64-s + 2·71-s − 72-s − 2·73-s + 81-s + 2·82-s − 92-s − 2·94-s − 98-s − 100-s + ⋯
L(s)  = 1  − 2-s + 4-s − 8-s + 9-s + 16-s − 18-s − 23-s − 25-s − 2·31-s − 32-s + 36-s − 2·41-s + 46-s + 2·47-s + 49-s + 50-s + 2·62-s + 64-s + 2·71-s − 72-s − 2·73-s + 81-s + 2·82-s − 92-s − 2·94-s − 98-s − 100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(184\)    =    \(2^{3} \cdot 23\)
Sign: $1$
Analytic conductor: \(0.0918279\)
Root analytic conductor: \(0.303031\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{184} (45, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 184,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4785633379\)
\(L(\frac12)\) \(\approx\) \(0.4785633379\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
23 \( 1 + T \)
good3 \( ( 1 - T )( 1 + T ) \)
5 \( 1 + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( 1 + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 + T )^{2} \)
37 \( 1 + T^{2} \)
41 \( ( 1 + T )^{2} \)
43 \( 1 + T^{2} \)
47 \( ( 1 - T )^{2} \)
53 \( 1 + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( ( 1 - T )^{2} \)
73 \( ( 1 + T )^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 + T^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−12.60382009645242516725282172874, −11.76890936112523724911841670715, −10.63124442421043472159737664036, −9.889475609096034897509971366092, −8.931820667411348977148642010353, −7.76069935185950185020083922704, −6.94933654383553911064683354634, −5.66526340069983104207980886428, −3.82563352360396967749654331175, −1.92065401555379753020071538316, 1.92065401555379753020071538316, 3.82563352360396967749654331175, 5.66526340069983104207980886428, 6.94933654383553911064683354634, 7.76069935185950185020083922704, 8.931820667411348977148642010353, 9.889475609096034897509971366092, 10.63124442421043472159737664036, 11.76890936112523724911841670715, 12.60382009645242516725282172874

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