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Properties

Label 2-1368-8.5-c1-0-7
Degree $2$
Conductor $1368$
Sign $-0.892 - 0.451i$
Analytic cond. $10.9235$
Root an. cond. $3.30507$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.09 − 0.889i)2-s + (0.418 + 1.95i)4-s + 3.13i·5-s − 0.535·7-s + (1.27 − 2.52i)8-s + (2.79 − 3.45i)10-s − 0.425i·11-s + 6.65i·13-s + (0.589 + 0.476i)14-s + (−3.64 + 1.63i)16-s − 7.33·17-s + i·19-s + (−6.13 + 1.31i)20-s + (−0.378 + 0.467i)22-s + 5.90·23-s + ⋯
L(s)  = 1  + (−0.777 − 0.628i)2-s + (0.209 + 0.977i)4-s + 1.40i·5-s − 0.202·7-s + (0.451 − 0.892i)8-s + (0.882 − 1.09i)10-s − 0.128i·11-s + 1.84i·13-s + (0.157 + 0.127i)14-s + (−0.912 + 0.409i)16-s − 1.77·17-s + 0.229i·19-s + (−1.37 + 0.293i)20-s + (−0.0806 + 0.0996i)22-s + 1.23·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.892 - 0.451i$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ -0.892 - 0.451i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4762352818\)
\(L(\frac12)\) \(\approx\) \(0.4762352818\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.09 + 0.889i)T \)
3 \( 1 \)
19 \( 1 - iT \)
good5 \( 1 - 3.13iT - 5T^{2} \)
7 \( 1 + 0.535T + 7T^{2} \)
11 \( 1 + 0.425iT - 11T^{2} \)
13 \( 1 - 6.65iT - 13T^{2} \)
17 \( 1 + 7.33T + 17T^{2} \)
23 \( 1 - 5.90T + 23T^{2} \)
29 \( 1 + 0.837iT - 29T^{2} \)
31 \( 1 + 3.16T + 31T^{2} \)
37 \( 1 + 3.49iT - 37T^{2} \)
41 \( 1 - 0.123T + 41T^{2} \)
43 \( 1 + 5.39iT - 43T^{2} \)
47 \( 1 - 2.02T + 47T^{2} \)
53 \( 1 + 5.82iT - 53T^{2} \)
59 \( 1 - 5.56iT - 59T^{2} \)
61 \( 1 + 6.99iT - 61T^{2} \)
67 \( 1 - 12.3iT - 67T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 + 6.99T + 73T^{2} \)
79 \( 1 - 2.07T + 79T^{2} \)
83 \( 1 + 11.8iT - 83T^{2} \)
89 \( 1 + 13.7T + 89T^{2} \)
97 \( 1 - 0.801T + 97T^{2} \)
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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

−9.977569829141676356332402785345, −9.108047182178687556265448372242, −8.676174084357280490509247835262, −7.25650924266034251977294210605, −6.96273473421729539410851951940, −6.24781822940852065405490299562, −4.52112999975662755205708266022, −3.67337905566778030657363740800, −2.64876652369390862941761420599, −1.85685422373047762021813650035, 0.25521068562842073471120673113, 1.37483643411427005758312948590, 2.83352481571519689325324349102, 4.52387232373040949787564987715, 5.12173788448613041673285764863, 5.94075623390733748895658697296, 6.91466930241707669977301525004, 7.81125860486686446337618098354, 8.557514448062141703527124361275, 9.031278020436071533296318741466

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