Properties

Label 2-468-1.1-c1-0-2
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 2·7-s + 4·11-s + 13-s − 2·17-s − 2·19-s + 11·25-s + 6·29-s − 10·31-s − 8·35-s + 10·37-s − 8·41-s + 4·43-s + 4·47-s − 3·49-s + 10·53-s + 16·55-s + 8·59-s − 14·61-s + 4·65-s + 2·67-s − 16·71-s − 10·73-s − 8·77-s − 16·79-s − 8·85-s + 4·89-s + ⋯
L(s)  = 1  + 1.78·5-s − 0.755·7-s + 1.20·11-s + 0.277·13-s − 0.485·17-s − 0.458·19-s + 11/5·25-s + 1.11·29-s − 1.79·31-s − 1.35·35-s + 1.64·37-s − 1.24·41-s + 0.609·43-s + 0.583·47-s − 3/7·49-s + 1.37·53-s + 2.15·55-s + 1.04·59-s − 1.79·61-s + 0.496·65-s + 0.244·67-s − 1.89·71-s − 1.17·73-s − 0.911·77-s − 1.80·79-s − 0.867·85-s + 0.423·89-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: $\chi_{468} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.826951888\)
\(L(\frac12)\) \(\approx\) \(1.826951888\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
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

−10.81609252174330783275568401880, −10.02637231667649988922232878626, −9.272613788586316089970782456314, −8.740006195634140431482743685580, −7.00743292844584939053022110799, −6.30352193131009441178367980896, −5.63885615575172245866908586348, −4.22765636287388069600769197082, −2.78430186696517884115720808997, −1.53061467325951463049581429920, 1.53061467325951463049581429920, 2.78430186696517884115720808997, 4.22765636287388069600769197082, 5.63885615575172245866908586348, 6.30352193131009441178367980896, 7.00743292844584939053022110799, 8.740006195634140431482743685580, 9.272613788586316089970782456314, 10.02637231667649988922232878626, 10.81609252174330783275568401880

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