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Properties

Label 2-1368-57.11-c2-0-39
Degree $2$
Conductor $1368$
Sign $0.162 - 0.986i$
Analytic cond. $37.2753$
Root an. cond. $6.10535$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.75 − 4.47i)5-s − 8.60·7-s − 12.8i·11-s + (−11.6 − 20.2i)13-s + (−22.1 − 12.7i)17-s + (−18.2 − 5.32i)19-s + (−0.190 + 0.110i)23-s + (27.5 + 47.7i)25-s + (33.8 − 19.5i)29-s + 10.5·31-s + (66.7 + 38.5i)35-s − 20.5·37-s + (−50.6 − 29.2i)41-s + (−24.2 + 41.9i)43-s + (71.7 − 41.4i)47-s + ⋯
L(s)  = 1  + (−1.55 − 0.895i)5-s − 1.22·7-s − 1.16i·11-s + (−0.897 − 1.55i)13-s + (−1.30 − 0.750i)17-s + (−0.959 − 0.280i)19-s + (−0.00829 + 0.00479i)23-s + (1.10 + 1.91i)25-s + (1.16 − 0.673i)29-s + 0.339·31-s + (1.90 + 1.10i)35-s − 0.554·37-s + (−1.23 − 0.713i)41-s + (−0.563 + 0.975i)43-s + (1.52 − 0.881i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.162 - 0.986i$
Analytic conductor: \(37.2753\)
Root analytic conductor: \(6.10535\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1),\ 0.162 - 0.986i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3306944770\)
\(L(\frac12)\) \(\approx\) \(0.3306944770\)
\(L(2)\) 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 \( 1 + (18.2 + 5.32i)T \)
good5 \( 1 + (7.75 + 4.47i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + 8.60T + 49T^{2} \)
11 \( 1 + 12.8iT - 121T^{2} \)
13 \( 1 + (11.6 + 20.2i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (22.1 + 12.7i)T + (144.5 + 250. i)T^{2} \)
23 \( 1 + (0.190 - 0.110i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-33.8 + 19.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 10.5T + 961T^{2} \)
37 \( 1 + 20.5T + 1.36e3T^{2} \)
41 \( 1 + (50.6 + 29.2i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (24.2 - 41.9i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-71.7 + 41.4i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-22.9 + 13.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (46.2 + 26.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (36.4 + 63.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (44.6 + 77.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-60.2 - 34.7i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (12.1 - 21.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (24.9 - 43.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 21.4iT - 6.88e3T^{2} \)
89 \( 1 + (45.6 - 26.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (46.2 - 80.1i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−8.548802143770620337060835205501, −8.209672299234619589327216312491, −7.18203542823617785471269573849, −6.40545424884888904567342658679, −5.25613343923786227881738503666, −4.46577138298725371162131320335, −3.47146523080651824942543275401, −2.75509118855654233459267895973, −0.43244516114965809898098321144, −0.22532131457850954113583426252, 2.10975991318699032648999905477, 3.09406864189512775127749294754, 4.23132574755923085705480539203, 4.47001204080442021205186612270, 6.37697544190983214088124089685, 6.87154021621279903062588455483, 7.27890029497133500299479678681, 8.457038158739993530955466060641, 9.153754933398978411052026531598, 10.22649830953773974073550628708

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