Properties

Label 2-1368-1.1-c3-0-20
Degree $2$
Conductor $1368$
Sign $1$
Analytic cond. $80.7146$
Root an. cond. $8.98413$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.38·5-s + 5.83·7-s + 7.33·11-s + 55.6·13-s + 10.0·17-s − 19·19-s + 9.26·23-s − 119.·25-s + 83.9·29-s + 202.·31-s − 13.9·35-s + 95.2·37-s + 25.9·41-s − 119.·43-s − 467.·47-s − 308.·49-s + 764.·53-s − 17.4·55-s − 69.1·59-s − 398.·61-s − 132.·65-s − 243.·67-s + 781.·71-s − 711.·73-s + 42.7·77-s + 723.·79-s + 1.22e3·83-s + ⋯
L(s)  = 1  − 0.213·5-s + 0.315·7-s + 0.201·11-s + 1.18·13-s + 0.143·17-s − 0.229·19-s + 0.0839·23-s − 0.954·25-s + 0.537·29-s + 1.17·31-s − 0.0671·35-s + 0.423·37-s + 0.0990·41-s − 0.424·43-s − 1.45·47-s − 0.900·49-s + 1.98·53-s − 0.0428·55-s − 0.152·59-s − 0.836·61-s − 0.253·65-s − 0.444·67-s + 1.30·71-s − 1.14·73-s + 0.0633·77-s + 1.03·79-s + 1.62·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(80.7146\)
Root analytic conductor: \(8.98413\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.261914381\)
\(L(\frac12)\) \(\approx\) \(2.261914381\)
\(L(\frac{5}{2})\) 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 \( 1 \)
19 \( 1 + 19T \)
good5 \( 1 + 2.38T + 125T^{2} \)
7 \( 1 - 5.83T + 343T^{2} \)
11 \( 1 - 7.33T + 1.33e3T^{2} \)
13 \( 1 - 55.6T + 2.19e3T^{2} \)
17 \( 1 - 10.0T + 4.91e3T^{2} \)
23 \( 1 - 9.26T + 1.21e4T^{2} \)
29 \( 1 - 83.9T + 2.43e4T^{2} \)
31 \( 1 - 202.T + 2.97e4T^{2} \)
37 \( 1 - 95.2T + 5.06e4T^{2} \)
41 \( 1 - 25.9T + 6.89e4T^{2} \)
43 \( 1 + 119.T + 7.95e4T^{2} \)
47 \( 1 + 467.T + 1.03e5T^{2} \)
53 \( 1 - 764.T + 1.48e5T^{2} \)
59 \( 1 + 69.1T + 2.05e5T^{2} \)
61 \( 1 + 398.T + 2.26e5T^{2} \)
67 \( 1 + 243.T + 3.00e5T^{2} \)
71 \( 1 - 781.T + 3.57e5T^{2} \)
73 \( 1 + 711.T + 3.89e5T^{2} \)
79 \( 1 - 723.T + 4.93e5T^{2} \)
83 \( 1 - 1.22e3T + 5.71e5T^{2} \)
89 \( 1 - 653.T + 7.04e5T^{2} \)
97 \( 1 - 1.69e3T + 9.12e5T^{2} \)
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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

−9.149836530724034358611818176582, −8.322486053659064466982726694266, −7.80311875964689259767438035785, −6.63778334734036382536019789928, −6.04031598578913490798043405643, −4.95325963562381644586389083996, −4.06239483658531536916375386736, −3.17457306963062186392959062198, −1.87732773814116378343467387801, −0.77270677029144829245397450576, 0.77270677029144829245397450576, 1.87732773814116378343467387801, 3.17457306963062186392959062198, 4.06239483658531536916375386736, 4.95325963562381644586389083996, 6.04031598578913490798043405643, 6.63778334734036382536019789928, 7.80311875964689259767438035785, 8.322486053659064466982726694266, 9.149836530724034358611818176582

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