Properties

Label 2-162288-1.1-c1-0-104
Degree $2$
Conductor $162288$
Sign $-1$
Analytic cond. $1295.87$
Root an. cond. $35.9982$
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  − 5-s + 2·11-s − 3·13-s + 2·19-s − 23-s − 4·25-s + 5·29-s + 37-s + 3·41-s + 7·43-s − 13·47-s + 6·53-s − 2·55-s + 12·59-s + 4·61-s + 3·65-s − 8·71-s − 2·73-s + 10·79-s + 4·83-s − 16·89-s − 2·95-s + 11·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.603·11-s − 0.832·13-s + 0.458·19-s − 0.208·23-s − 4/5·25-s + 0.928·29-s + 0.164·37-s + 0.468·41-s + 1.06·43-s − 1.89·47-s + 0.824·53-s − 0.269·55-s + 1.56·59-s + 0.512·61-s + 0.372·65-s − 0.949·71-s − 0.234·73-s + 1.12·79-s + 0.439·83-s − 1.69·89-s − 0.205·95-s + 1.11·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162288\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(1295.87\)
Root analytic conductor: \(35.9982\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{162288} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 162288,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
7 \( 1 \)
23 \( 1 + T \)
good5 \( 1 + T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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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

−13.44174077075362, −13.04669674051055, −12.46267476944766, −12.00102135748537, −11.65589534087746, −11.31607489077618, −10.61275754958513, −10.10201234961684, −9.712871773771553, −9.243712501805175, −8.694238589241738, −8.089844443843148, −7.778699565749055, −7.149773545889401, −6.768600138246088, −6.169349543815310, −5.616098441390246, −5.036034229803082, −4.516733403577945, −3.941593142228905, −3.531238979340714, −2.729038402054537, −2.304373584449285, −1.475402592212029, −0.8040435085352862, 0, 0.8040435085352862, 1.475402592212029, 2.304373584449285, 2.729038402054537, 3.531238979340714, 3.941593142228905, 4.516733403577945, 5.036034229803082, 5.616098441390246, 6.169349543815310, 6.768600138246088, 7.149773545889401, 7.778699565749055, 8.089844443843148, 8.694238589241738, 9.243712501805175, 9.712871773771553, 10.10201234961684, 10.61275754958513, 11.31607489077618, 11.65589534087746, 12.00102135748537, 12.46267476944766, 13.04669674051055, 13.44174077075362

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