Properties

Label 2-16184-1.1-c1-0-4
Degree $2$
Conductor $16184$
Sign $-1$
Analytic cond. $129.229$
Root an. cond. $11.3679$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 2·5-s + 7-s + 9-s − 4·13-s + 4·15-s − 6·19-s + 2·21-s + 3·23-s − 25-s − 4·27-s − 5·29-s − 2·31-s + 2·35-s + 3·37-s − 8·39-s + 7·43-s + 2·45-s + 49-s − 53-s − 12·57-s − 14·59-s − 2·61-s + 63-s − 8·65-s + 4·67-s + 6·69-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.10·13-s + 1.03·15-s − 1.37·19-s + 0.436·21-s + 0.625·23-s − 1/5·25-s − 0.769·27-s − 0.928·29-s − 0.359·31-s + 0.338·35-s + 0.493·37-s − 1.28·39-s + 1.06·43-s + 0.298·45-s + 1/7·49-s − 0.137·53-s − 1.58·57-s − 1.82·59-s − 0.256·61-s + 0.125·63-s − 0.992·65-s + 0.488·67-s + 0.722·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16184\)    =    \(2^{3} \cdot 7 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(129.229\)
Root analytic conductor: \(11.3679\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{16184} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 16184,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
17 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 11 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−16.25067082984995, −15.30887864968814, −15.06341209054441, −14.42089355793226, −14.22799572756187, −13.50177160384999, −13.00441779552708, −12.58500000924708, −11.77881390917316, −11.07278274288795, −10.51835544632158, −9.839982563692878, −9.304126721785830, −8.935198351344527, −8.289761583117375, −7.555829306176327, −7.243072051715566, −6.187647672101114, −5.777959124239697, −4.887100829797342, −4.303584199712500, −3.458863328831882, −2.609836530048954, −2.198446929618027, −1.511630529097296, 0, 1.511630529097296, 2.198446929618027, 2.609836530048954, 3.458863328831882, 4.303584199712500, 4.887100829797342, 5.777959124239697, 6.187647672101114, 7.243072051715566, 7.555829306176327, 8.289761583117375, 8.935198351344527, 9.304126721785830, 9.839982563692878, 10.51835544632158, 11.07278274288795, 11.77881390917316, 12.58500000924708, 13.00441779552708, 13.50177160384999, 14.22799572756187, 14.42089355793226, 15.06341209054441, 15.30887864968814, 16.25067082984995

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