Properties

Label 2-1368-1.1-c3-0-19
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  + 13.8·5-s − 9.22·7-s − 59.8·11-s + 38.2·13-s + 47.0·17-s − 19·19-s − 179.·23-s + 66.1·25-s + 96.6·29-s + 125.·31-s − 127.·35-s + 407.·37-s + 220.·41-s + 100.·43-s + 213.·47-s − 257.·49-s − 19.8·53-s − 827.·55-s + 97.7·59-s + 266.·61-s + 528.·65-s + 627.·67-s + 644.·71-s + 807.·73-s + 552.·77-s + 785.·79-s − 1.17e3·83-s + ⋯
L(s)  = 1  + 1.23·5-s − 0.497·7-s − 1.64·11-s + 0.815·13-s + 0.671·17-s − 0.229·19-s − 1.62·23-s + 0.529·25-s + 0.618·29-s + 0.729·31-s − 0.615·35-s + 1.81·37-s + 0.840·41-s + 0.357·43-s + 0.663·47-s − 0.752·49-s − 0.0513·53-s − 2.02·55-s + 0.215·59-s + 0.559·61-s + 1.00·65-s + 1.14·67-s + 1.07·71-s + 1.29·73-s + 0.817·77-s + 1.11·79-s − 1.55·83-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

\(L(2)\) \(\approx\) \(2.346241156\)
\(L(\frac12)\) \(\approx\) \(2.346241156\)
\(L(\frac{5}{2})\) 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 \( 1 - 13.8T + 125T^{2} \)
7 \( 1 + 9.22T + 343T^{2} \)
11 \( 1 + 59.8T + 1.33e3T^{2} \)
13 \( 1 - 38.2T + 2.19e3T^{2} \)
17 \( 1 - 47.0T + 4.91e3T^{2} \)
23 \( 1 + 179.T + 1.21e4T^{2} \)
29 \( 1 - 96.6T + 2.43e4T^{2} \)
31 \( 1 - 125.T + 2.97e4T^{2} \)
37 \( 1 - 407.T + 5.06e4T^{2} \)
41 \( 1 - 220.T + 6.89e4T^{2} \)
43 \( 1 - 100.T + 7.95e4T^{2} \)
47 \( 1 - 213.T + 1.03e5T^{2} \)
53 \( 1 + 19.8T + 1.48e5T^{2} \)
59 \( 1 - 97.7T + 2.05e5T^{2} \)
61 \( 1 - 266.T + 2.26e5T^{2} \)
67 \( 1 - 627.T + 3.00e5T^{2} \)
71 \( 1 - 644.T + 3.57e5T^{2} \)
73 \( 1 - 807.T + 3.89e5T^{2} \)
79 \( 1 - 785.T + 4.93e5T^{2} \)
83 \( 1 + 1.17e3T + 5.71e5T^{2} \)
89 \( 1 - 234.T + 7.04e5T^{2} \)
97 \( 1 + 1.34e3T + 9.12e5T^{2} \)
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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

−9.500143484924736376280989621112, −8.278234953368324692191596441840, −7.81007820441396723987890900230, −6.48540704370522943084394513308, −5.92782654503080430395915149829, −5.27087304134275618165960456498, −4.07626648596919512053100121966, −2.83237459269324859353575245478, −2.13104078861082743545416752178, −0.75273871135936216228170881703, 0.75273871135936216228170881703, 2.13104078861082743545416752178, 2.83237459269324859353575245478, 4.07626648596919512053100121966, 5.27087304134275618165960456498, 5.92782654503080430395915149829, 6.48540704370522943084394513308, 7.81007820441396723987890900230, 8.278234953368324692191596441840, 9.500143484924736376280989621112

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