Properties

Label 2-1368-1.1-c3-0-1
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  − 18.4·5-s + 10.3·7-s − 50.4·11-s − 61.8·13-s − 68.1·17-s − 19·19-s − 145.·23-s + 215.·25-s − 42.6·29-s + 91.6·31-s − 191.·35-s − 400.·37-s + 123.·41-s + 449.·43-s + 453.·47-s − 235.·49-s − 437.·53-s + 930.·55-s + 159.·59-s − 476.·61-s + 1.14e3·65-s − 629.·67-s − 471.·71-s − 725.·73-s − 524.·77-s − 1.05e3·79-s + 726.·83-s + ⋯
L(s)  = 1  − 1.64·5-s + 0.560·7-s − 1.38·11-s − 1.31·13-s − 0.971·17-s − 0.229·19-s − 1.32·23-s + 1.72·25-s − 0.272·29-s + 0.531·31-s − 0.925·35-s − 1.78·37-s + 0.469·41-s + 1.59·43-s + 1.40·47-s − 0.685·49-s − 1.13·53-s + 2.28·55-s + 0.351·59-s − 1.00·61-s + 2.17·65-s − 1.14·67-s − 0.788·71-s − 1.16·73-s − 0.775·77-s − 1.50·79-s + 0.961·83-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

\(L(2)\) \(\approx\) \(0.3333365258\)
\(L(\frac12)\) \(\approx\) \(0.3333365258\)
\(L(\frac{5}{2})\) 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 \( 1 + 18.4T + 125T^{2} \)
7 \( 1 - 10.3T + 343T^{2} \)
11 \( 1 + 50.4T + 1.33e3T^{2} \)
13 \( 1 + 61.8T + 2.19e3T^{2} \)
17 \( 1 + 68.1T + 4.91e3T^{2} \)
23 \( 1 + 145.T + 1.21e4T^{2} \)
29 \( 1 + 42.6T + 2.43e4T^{2} \)
31 \( 1 - 91.6T + 2.97e4T^{2} \)
37 \( 1 + 400.T + 5.06e4T^{2} \)
41 \( 1 - 123.T + 6.89e4T^{2} \)
43 \( 1 - 449.T + 7.95e4T^{2} \)
47 \( 1 - 453.T + 1.03e5T^{2} \)
53 \( 1 + 437.T + 1.48e5T^{2} \)
59 \( 1 - 159.T + 2.05e5T^{2} \)
61 \( 1 + 476.T + 2.26e5T^{2} \)
67 \( 1 + 629.T + 3.00e5T^{2} \)
71 \( 1 + 471.T + 3.57e5T^{2} \)
73 \( 1 + 725.T + 3.89e5T^{2} \)
79 \( 1 + 1.05e3T + 4.93e5T^{2} \)
83 \( 1 - 726.T + 5.71e5T^{2} \)
89 \( 1 - 468.T + 7.04e5T^{2} \)
97 \( 1 + 891.T + 9.12e5T^{2} \)
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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

−9.018138688923631320908941770084, −8.223834607992527798035250090443, −7.60165612502379080435796559388, −7.18624896997145261462184048214, −5.81141440956033451756835764575, −4.69757155332745383256131538028, −4.32713675282019254256712809019, −3.07627925548040200749505196756, −2.10507117737632768662242998237, −0.26945740142061255147756475104, 0.26945740142061255147756475104, 2.10507117737632768662242998237, 3.07627925548040200749505196756, 4.32713675282019254256712809019, 4.69757155332745383256131538028, 5.81141440956033451756835764575, 7.18624896997145261462184048214, 7.60165612502379080435796559388, 8.223834607992527798035250090443, 9.018138688923631320908941770084

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