Properties

Label 2-3312-1.1-c1-0-27
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.64·5-s − 3.17·7-s + 0.962·11-s + 5.85·13-s − 2.68·17-s + 5.50·19-s − 23-s + 8.29·25-s + 1.85·29-s + 1.85·31-s + 11.5·35-s − 6.67·37-s + 9.14·41-s − 7.57·43-s − 8.34·47-s + 3.06·49-s + 5.99·53-s − 3.50·55-s − 8.80·59-s + 1.03·61-s − 21.3·65-s + 1.08·67-s − 13.1·71-s + 13.2·73-s − 3.05·77-s + 4.53·79-s − 3.03·83-s + ⋯
L(s)  = 1  − 1.63·5-s − 1.19·7-s + 0.290·11-s + 1.62·13-s − 0.650·17-s + 1.26·19-s − 0.208·23-s + 1.65·25-s + 0.344·29-s + 0.333·31-s + 1.95·35-s − 1.09·37-s + 1.42·41-s − 1.15·43-s − 1.21·47-s + 0.438·49-s + 0.823·53-s − 0.473·55-s − 1.14·59-s + 0.132·61-s − 2.64·65-s + 0.132·67-s − 1.56·71-s + 1.54·73-s − 0.348·77-s + 0.510·79-s − 0.333·83-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: \(1\)
Selberg data: \((2,\ 3312,\ (\ :1/2),\ -1)\)

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 \( 1 \)
23 \( 1 + T \)
good5 \( 1 + 3.64T + 5T^{2} \)
7 \( 1 + 3.17T + 7T^{2} \)
11 \( 1 - 0.962T + 11T^{2} \)
13 \( 1 - 5.85T + 13T^{2} \)
17 \( 1 + 2.68T + 17T^{2} \)
19 \( 1 - 5.50T + 19T^{2} \)
29 \( 1 - 1.85T + 29T^{2} \)
31 \( 1 - 1.85T + 31T^{2} \)
37 \( 1 + 6.67T + 37T^{2} \)
41 \( 1 - 9.14T + 41T^{2} \)
43 \( 1 + 7.57T + 43T^{2} \)
47 \( 1 + 8.34T + 47T^{2} \)
53 \( 1 - 5.99T + 53T^{2} \)
59 \( 1 + 8.80T + 59T^{2} \)
61 \( 1 - 1.03T + 61T^{2} \)
67 \( 1 - 1.08T + 67T^{2} \)
71 \( 1 + 13.1T + 71T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 - 4.53T + 79T^{2} \)
83 \( 1 + 3.03T + 83T^{2} \)
89 \( 1 - 3.10T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−8.260019894338825306890317397785, −7.55941621745713018641895081039, −6.71717736306354847153733575192, −6.26034909508807615617650401330, −5.12906385454612032054815384106, −4.06868891797785005378748187966, −3.58358143401514259989907936739, −2.94156428202064289019780531163, −1.19313696608815500155426813436, 0, 1.19313696608815500155426813436, 2.94156428202064289019780531163, 3.58358143401514259989907936739, 4.06868891797785005378748187966, 5.12906385454612032054815384106, 6.26034909508807615617650401330, 6.71717736306354847153733575192, 7.55941621745713018641895081039, 8.260019894338825306890317397785

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