Properties

Label 2-16184-1.1-c1-0-5
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 − 2·11-s − 2·13-s + 4·15-s + 4·19-s + 2·21-s − 4·23-s − 25-s − 4·27-s + 2·29-s − 8·31-s − 4·33-s + 2·35-s − 2·37-s − 4·39-s + 4·43-s + 2·45-s − 12·47-s + 49-s − 14·53-s − 4·55-s + 8·57-s + 12·59-s − 10·61-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 1.03·15-s + 0.917·19-s + 0.436·21-s − 0.834·23-s − 1/5·25-s − 0.769·27-s + 0.371·29-s − 1.43·31-s − 0.696·33-s + 0.338·35-s − 0.328·37-s − 0.640·39-s + 0.609·43-s + 0.298·45-s − 1.75·47-s + 1/7·49-s − 1.92·53-s − 0.539·55-s + 1.05·57-s + 1.56·59-s − 1.28·61-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−15.95085839460561, −15.79738493279583, −14.84933388546332, −14.43684835264432, −14.16260908728535, −13.53276465693366, −13.09420039518025, −12.52044954819450, −11.71661270545571, −11.21199422563370, −10.37385892964625, −9.858576864737306, −9.445758159630633, −8.859199547326198, −8.185676681078533, −7.688117839729930, −7.207510358903617, −6.238452508806355, −5.606158833987239, −5.072209588295550, −4.241885898654914, −3.350182325848894, −2.821269115000839, −2.027884073349522, −1.567926273354090, 0, 1.567926273354090, 2.027884073349522, 2.821269115000839, 3.350182325848894, 4.241885898654914, 5.072209588295550, 5.606158833987239, 6.238452508806355, 7.207510358903617, 7.688117839729930, 8.185676681078533, 8.859199547326198, 9.445758159630633, 9.858576864737306, 10.37385892964625, 11.21199422563370, 11.71661270545571, 12.52044954819450, 13.09420039518025, 13.53276465693366, 14.16260908728535, 14.43684835264432, 14.84933388546332, 15.79738493279583, 15.95085839460561

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