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Properties

Label 2-3312-1.1-c1-0-48
Degree $2$
Conductor $3312$
Sign $-1$
Analytic cond. $26.4464$
Root an. cond. $5.14261$
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  + 2·7-s − 5·13-s + 6·17-s − 6·19-s + 23-s − 5·25-s − 9·29-s − 3·31-s − 8·37-s − 3·41-s + 8·43-s + 7·47-s − 3·49-s + 2·53-s + 4·59-s − 10·61-s − 8·67-s + 7·71-s + 9·73-s + 6·79-s − 14·83-s − 16·89-s − 10·91-s + 6·97-s − 6·101-s − 14·103-s + 14·107-s + ⋯
L(s)  = 1  + 0.755·7-s − 1.38·13-s + 1.45·17-s − 1.37·19-s + 0.208·23-s − 25-s − 1.67·29-s − 0.538·31-s − 1.31·37-s − 0.468·41-s + 1.21·43-s + 1.02·47-s − 3/7·49-s + 0.274·53-s + 0.520·59-s − 1.28·61-s − 0.977·67-s + 0.830·71-s + 1.05·73-s + 0.675·79-s − 1.53·83-s − 1.69·89-s − 1.04·91-s + 0.609·97-s − 0.597·101-s − 1.37·103-s + 1.35·107-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: yes
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 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−8.110478443981525229338745686859, −7.57574393299778347582875795759, −6.96311085153201898297894475156, −5.77320141887774085455693828185, −5.30781951251755615786230141863, −4.38617246092236815909135983678, −3.59004246894799179150414569661, −2.40410911892081301411643930480, −1.61191613063293050436333394676, 0, 1.61191613063293050436333394676, 2.40410911892081301411643930480, 3.59004246894799179150414569661, 4.38617246092236815909135983678, 5.30781951251755615786230141863, 5.77320141887774085455693828185, 6.96311085153201898297894475156, 7.57574393299778347582875795759, 8.110478443981525229338745686859

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