Properties

Label 2-468-1.1-c3-0-11
Degree $2$
Conductor $468$
Sign $-1$
Analytic cond. $27.6128$
Root an. cond. $5.25479$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 32·7-s + 68·11-s + 13·13-s + 14·17-s + 4·19-s − 72·23-s − 121·25-s − 102·29-s − 136·31-s − 64·35-s − 386·37-s − 250·41-s − 140·43-s + 296·47-s + 681·49-s − 526·53-s + 136·55-s − 332·59-s − 410·61-s + 26·65-s + 596·67-s + 880·71-s + 506·73-s − 2.17e3·77-s − 640·79-s − 1.38e3·83-s + ⋯
L(s)  = 1  + 0.178·5-s − 1.72·7-s + 1.86·11-s + 0.277·13-s + 0.199·17-s + 0.0482·19-s − 0.652·23-s − 0.967·25-s − 0.653·29-s − 0.787·31-s − 0.309·35-s − 1.71·37-s − 0.952·41-s − 0.496·43-s + 0.918·47-s + 1.98·49-s − 1.36·53-s + 0.333·55-s − 0.732·59-s − 0.860·61-s + 0.0496·65-s + 1.08·67-s + 1.47·71-s + 0.811·73-s − 3.22·77-s − 0.911·79-s − 1.82·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(27.6128\)
Root analytic conductor: \(5.25479\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 468,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
13 \( 1 - p T \)
good5 \( 1 - 2 T + p^{3} T^{2} \)
7 \( 1 + 32 T + p^{3} T^{2} \)
11 \( 1 - 68 T + p^{3} T^{2} \)
17 \( 1 - 14 T + p^{3} T^{2} \)
19 \( 1 - 4 T + p^{3} T^{2} \)
23 \( 1 + 72 T + p^{3} T^{2} \)
29 \( 1 + 102 T + p^{3} T^{2} \)
31 \( 1 + 136 T + p^{3} T^{2} \)
37 \( 1 + 386 T + p^{3} T^{2} \)
41 \( 1 + 250 T + p^{3} T^{2} \)
43 \( 1 + 140 T + p^{3} T^{2} \)
47 \( 1 - 296 T + p^{3} T^{2} \)
53 \( 1 + 526 T + p^{3} T^{2} \)
59 \( 1 + 332 T + p^{3} T^{2} \)
61 \( 1 + 410 T + p^{3} T^{2} \)
67 \( 1 - 596 T + p^{3} T^{2} \)
71 \( 1 - 880 T + p^{3} T^{2} \)
73 \( 1 - 506 T + p^{3} T^{2} \)
79 \( 1 + 640 T + p^{3} T^{2} \)
83 \( 1 + 1380 T + p^{3} T^{2} \)
89 \( 1 + 1450 T + p^{3} T^{2} \)
97 \( 1 + 446 T + p^{3} T^{2} \)
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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

−9.854874963591984556567134082862, −9.476546513981004135339123064073, −8.579754041476501043358351941633, −7.12115902380027228718903012808, −6.45933825527282320912357798157, −5.68120483184912008031639667316, −3.96021769021079225511584685798, −3.35692263862781988931026014923, −1.67059934989288439502634804305, 0, 1.67059934989288439502634804305, 3.35692263862781988931026014923, 3.96021769021079225511584685798, 5.68120483184912008031639667316, 6.45933825527282320912357798157, 7.12115902380027228718903012808, 8.579754041476501043358351941633, 9.476546513981004135339123064073, 9.854874963591984556567134082862

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