Properties

Label 2-3312-1.1-c1-0-13
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  − 2·5-s + 5.12·11-s + 4.56·13-s + 3.12·17-s − 5.12·19-s − 23-s − 25-s + 0.561·29-s + 6.56·31-s − 8.24·37-s − 10.8·41-s + 8·43-s + 11.6·47-s − 7·49-s − 2·53-s − 10.2·55-s − 6.24·59-s + 12.2·61-s − 9.12·65-s + 5.12·67-s + 9.43·71-s − 2.31·73-s + 5.12·79-s − 2.24·83-s − 6.24·85-s + 13.3·89-s + 10.2·95-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.54·11-s + 1.26·13-s + 0.757·17-s − 1.17·19-s − 0.208·23-s − 0.200·25-s + 0.104·29-s + 1.17·31-s − 1.35·37-s − 1.68·41-s + 1.21·43-s + 1.70·47-s − 49-s − 0.274·53-s − 1.38·55-s − 0.813·59-s + 1.56·61-s − 1.13·65-s + 0.625·67-s + 1.12·71-s − 0.270·73-s + 0.576·79-s − 0.246·83-s − 0.677·85-s + 1.41·89-s + 1.05·95-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: $\chi_{3312} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3312,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.763241098\)
\(L(\frac12)\) \(\approx\) \(1.763241098\)
\(L(\frac{3}{2})\) 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 \( 1 + 2T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 5.12T + 11T^{2} \)
13 \( 1 - 4.56T + 13T^{2} \)
17 \( 1 - 3.12T + 17T^{2} \)
19 \( 1 + 5.12T + 19T^{2} \)
29 \( 1 - 0.561T + 29T^{2} \)
31 \( 1 - 6.56T + 31T^{2} \)
37 \( 1 + 8.24T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 6.24T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 - 5.12T + 67T^{2} \)
71 \( 1 - 9.43T + 71T^{2} \)
73 \( 1 + 2.31T + 73T^{2} \)
79 \( 1 - 5.12T + 79T^{2} \)
83 \( 1 + 2.24T + 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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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

−8.484652865828331613722270960161, −8.116282381550409061528746723874, −7.03716308732209282300110852364, −6.46915736737043775846941667305, −5.75026148095619163971865868535, −4.56005859389813946471320411952, −3.85227798450954084882438326413, −3.38799005375643845830239346399, −1.89323272175238904203744384988, −0.826726240466976182437078614861, 0.826726240466976182437078614861, 1.89323272175238904203744384988, 3.38799005375643845830239346399, 3.85227798450954084882438326413, 4.56005859389813946471320411952, 5.75026148095619163971865868535, 6.46915736737043775846941667305, 7.03716308732209282300110852364, 8.116282381550409061528746723874, 8.484652865828331613722270960161

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