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Properties

Label 2-184-184.45-c0-0-1
Degree $2$
Conductor $184$
Sign $1$
Analytic cond. $0.0918279$
Root an. cond. $0.303031$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + 1.73i·3-s + (−0.499 − 0.866i)4-s + (1.49 + 0.866i)6-s − 0.999·8-s − 1.99·9-s + (1.49 − 0.866i)12-s − 1.73i·13-s + (−0.5 + 0.866i)16-s + (−0.999 + 1.73i)18-s − 23-s − 1.73i·24-s − 25-s + (−1.49 − 0.866i)26-s − 1.73i·27-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + 1.73i·3-s + (−0.499 − 0.866i)4-s + (1.49 + 0.866i)6-s − 0.999·8-s − 1.99·9-s + (1.49 − 0.866i)12-s − 1.73i·13-s + (−0.5 + 0.866i)16-s + (−0.999 + 1.73i)18-s − 23-s − 1.73i·24-s − 25-s + (−1.49 − 0.866i)26-s − 1.73i·27-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(184\)    =    \(2^{3} \cdot 23\)
Sign: $1$
Analytic conductor: \(0.0918279\)
Root analytic conductor: \(0.303031\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{184} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 184,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7805875715\)
\(L(\frac12)\) \(\approx\) \(0.7805875715\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + T \)
good3 \( 1 - 1.73iT - T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + 1.73iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
29 \( 1 - 1.73iT - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
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

−12.73450827178459867040573226024, −11.68149406203610723558258075500, −10.64299439422787121241737197914, −10.20868582024761209641104733360, −9.319536142440528921527770208754, −8.206175263689184412407679820488, −5.88206029672434117157282978520, −5.06483423795029329512520436940, −3.93292280377889332414245401166, −2.94719270292561927370400946891, 2.25198209300471212712706150231, 4.21028620489519597444489717303, 5.94437251995716875291301790728, 6.57311791696696270035058239893, 7.56162980886532190479026971444, 8.301570214589871528252209263656, 9.488495768896575088723318240994, 11.69243299869226911288025580321, 11.94239805290365940957762084181, 13.10995043838533892545172818669

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