Properties

Label 2-468-1.1-c1-0-4
Degree $2$
Conductor $468$
Sign $-1$
Analytic cond. $3.73699$
Root an. cond. $1.93313$
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  − 2·5-s − 2·7-s + 2·11-s − 13-s − 6·17-s − 6·19-s − 8·23-s − 25-s − 2·29-s + 10·31-s + 4·35-s − 6·37-s + 6·41-s + 4·43-s + 2·47-s − 3·49-s − 6·53-s − 4·55-s + 10·59-s − 2·61-s + 2·65-s + 10·67-s − 10·71-s + 2·73-s − 4·77-s − 4·79-s + 6·83-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.755·7-s + 0.603·11-s − 0.277·13-s − 1.45·17-s − 1.37·19-s − 1.66·23-s − 1/5·25-s − 0.371·29-s + 1.79·31-s + 0.676·35-s − 0.986·37-s + 0.937·41-s + 0.609·43-s + 0.291·47-s − 3/7·49-s − 0.824·53-s − 0.539·55-s + 1.30·59-s − 0.256·61-s + 0.248·65-s + 1.22·67-s − 1.18·71-s + 0.234·73-s − 0.455·77-s − 0.450·79-s + 0.658·83-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(3.73699\)
Root analytic conductor: \(1.93313\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 468,\ (\ :1/2),\ -1)\)

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

−10.61844388374487874396678510357, −9.692364091479303392864086465395, −8.721135130247514032619362683743, −7.929338724462847919707075122995, −6.76526090609767395021650319394, −6.13695159170847990530021082389, −4.46186730170941536615099460323, −3.82242662960250469039719313923, −2.31233397883866071992439487427, 0, 2.31233397883866071992439487427, 3.82242662960250469039719313923, 4.46186730170941536615099460323, 6.13695159170847990530021082389, 6.76526090609767395021650319394, 7.929338724462847919707075122995, 8.721135130247514032619362683743, 9.692364091479303392864086465395, 10.61844388374487874396678510357

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