Properties

Label 2-3312-1.1-c1-0-33
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  − 1.64·5-s − 2·7-s − 1.64·11-s + 5.29·13-s + 3.29·17-s − 0.354·19-s + 23-s − 2.29·25-s − 9.29·29-s + 1.29·31-s + 3.29·35-s + 6.93·37-s − 6·41-s − 0.354·43-s + 6·47-s − 3·49-s + 1.64·53-s + 2.70·55-s + 0.354·61-s − 8.70·65-s + 14.9·67-s − 6·71-s − 7.29·73-s + 3.29·77-s − 8.58·79-s − 13.6·83-s − 5.41·85-s + ⋯
L(s)  = 1  − 0.736·5-s − 0.755·7-s − 0.496·11-s + 1.46·13-s + 0.798·17-s − 0.0812·19-s + 0.208·23-s − 0.458·25-s − 1.72·29-s + 0.231·31-s + 0.556·35-s + 1.14·37-s − 0.937·41-s − 0.0540·43-s + 0.875·47-s − 0.428·49-s + 0.226·53-s + 0.365·55-s + 0.0453·61-s − 1.08·65-s + 1.82·67-s − 0.712·71-s − 0.853·73-s + 0.375·77-s − 0.965·79-s − 1.49·83-s − 0.587·85-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: \(1\)
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 \( 1 + 1.64T + 5T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + 1.64T + 11T^{2} \)
13 \( 1 - 5.29T + 13T^{2} \)
17 \( 1 - 3.29T + 17T^{2} \)
19 \( 1 + 0.354T + 19T^{2} \)
29 \( 1 + 9.29T + 29T^{2} \)
31 \( 1 - 1.29T + 31T^{2} \)
37 \( 1 - 6.93T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 0.354T + 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 1.64T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 0.354T + 61T^{2} \)
67 \( 1 - 14.9T + 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 7.29T + 73T^{2} \)
79 \( 1 + 8.58T + 79T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 1.29T + 97T^{2} \)
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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

−8.172414220195013807580683290242, −7.62666819220307704120052098623, −6.79034758958093079187862454044, −5.95982348637053460917793896809, −5.36486406890763205413012073007, −4.07894025281306770148853689478, −3.63190438587214458613911508222, −2.73974777453234279774700692214, −1.34484504523455491022636950684, 0, 1.34484504523455491022636950684, 2.73974777453234279774700692214, 3.63190438587214458613911508222, 4.07894025281306770148853689478, 5.36486406890763205413012073007, 5.95982348637053460917793896809, 6.79034758958093079187862454044, 7.62666819220307704120052098623, 8.172414220195013807580683290242

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