Properties

Label 2-3312-1.1-c1-0-20
Degree $2$
Conductor $3312$
Sign $1$
Analytic cond. $26.4464$
Root an. cond. $5.14261$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.23·5-s + 1.23·7-s − 0.763·11-s + 3·13-s − 5.23·17-s + 2·19-s + 23-s + 5.47·25-s + 3·29-s + 6.70·31-s + 4.00·35-s + 3.23·37-s − 5.47·41-s + 2.23·47-s − 5.47·49-s + 8.47·53-s − 2.47·55-s − 2.47·59-s + 10.9·61-s + 9.70·65-s + 7.23·67-s + 7.76·71-s + 15.4·73-s − 0.944·77-s − 6.94·79-s − 13.2·83-s − 16.9·85-s + ⋯
L(s)  = 1  + 1.44·5-s + 0.467·7-s − 0.230·11-s + 0.832·13-s − 1.26·17-s + 0.458·19-s + 0.208·23-s + 1.09·25-s + 0.557·29-s + 1.20·31-s + 0.676·35-s + 0.532·37-s − 0.854·41-s + 0.326·47-s − 0.781·49-s + 1.16·53-s − 0.333·55-s − 0.321·59-s + 1.40·61-s + 1.20·65-s + 0.884·67-s + 0.921·71-s + 1.81·73-s − 0.107·77-s − 0.781·79-s − 1.45·83-s − 1.83·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3312\)    =    \(2^{4} \cdot 3^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(26.4464\)
Root analytic conductor: \(5.14261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3312,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.772695111\)
\(L(\frac12)\) \(\approx\) \(2.772695111\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23 \( 1 - T \)
good5 \( 1 - 3.23T + 5T^{2} \)
7 \( 1 - 1.23T + 7T^{2} \)
11 \( 1 + 0.763T + 11T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + 5.23T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 - 6.70T + 31T^{2} \)
37 \( 1 - 3.23T + 37T^{2} \)
41 \( 1 + 5.47T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 2.23T + 47T^{2} \)
53 \( 1 - 8.47T + 53T^{2} \)
59 \( 1 + 2.47T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 - 7.23T + 67T^{2} \)
71 \( 1 - 7.76T + 71T^{2} \)
73 \( 1 - 15.4T + 73T^{2} \)
79 \( 1 + 6.94T + 79T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 - 1.52T + 89T^{2} \)
97 \( 1 - 4.29T + 97T^{2} \)
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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

−8.591534390928548757234011838919, −8.129268070565906120717895357667, −6.85858974195832455577073939946, −6.44479739570541910052410123862, −5.57640241803011706020389774711, −4.96933744408047972299717843674, −4.03226673433012479853632874276, −2.79410498399196885244730163888, −2.05175877783164207980030865695, −1.06212686368271301802238229240, 1.06212686368271301802238229240, 2.05175877783164207980030865695, 2.79410498399196885244730163888, 4.03226673433012479853632874276, 4.96933744408047972299717843674, 5.57640241803011706020389774711, 6.44479739570541910052410123862, 6.85858974195832455577073939946, 8.129268070565906120717895357667, 8.591534390928548757234011838919

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