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Properties

Label 2-387-129.128-c1-0-10
Degree $2$
Conductor $387$
Sign $-0.538 + 0.842i$
Analytic cond. $3.09021$
Root an. cond. $1.75789$
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  − 2.69·2-s + 5.24·4-s + 1.42·5-s − 4.28i·7-s − 8.74·8-s − 3.83·10-s − 3.75i·11-s − 4.77·13-s + 11.5i·14-s + 13.0·16-s − 3.84i·17-s + 6.29i·19-s + 7.48·20-s + 10.1i·22-s + 1.76i·23-s + ⋯
L(s)  = 1  − 1.90·2-s + 2.62·4-s + 0.637·5-s − 1.61i·7-s − 3.09·8-s − 1.21·10-s − 1.13i·11-s − 1.32·13-s + 3.08i·14-s + 3.26·16-s − 0.932i·17-s + 1.44i·19-s + 1.67·20-s + 2.15i·22-s + 0.368i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $-0.538 + 0.842i$
Analytic conductor: \(3.09021\)
Root analytic conductor: \(1.75789\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (386, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :1/2),\ -0.538 + 0.842i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.211735 - 0.386755i\)
\(L(\frac12)\) \(\approx\) \(0.211735 - 0.386755i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 + (2.47 + 6.07i)T \)
good2 \( 1 + 2.69T + 2T^{2} \)
5 \( 1 - 1.42T + 5T^{2} \)
7 \( 1 + 4.28iT - 7T^{2} \)
11 \( 1 + 3.75iT - 11T^{2} \)
13 \( 1 + 4.77T + 13T^{2} \)
17 \( 1 + 3.84iT - 17T^{2} \)
19 \( 1 - 6.29iT - 19T^{2} \)
23 \( 1 - 1.76iT - 23T^{2} \)
29 \( 1 + 4.63T + 29T^{2} \)
31 \( 1 + 0.941T + 31T^{2} \)
37 \( 1 + 5.59iT - 37T^{2} \)
41 \( 1 - 2.16iT - 41T^{2} \)
47 \( 1 - 2.77iT - 47T^{2} \)
53 \( 1 - 1.18iT - 53T^{2} \)
59 \( 1 + 7.58iT - 59T^{2} \)
61 \( 1 + 9.63iT - 61T^{2} \)
67 \( 1 - 15.3T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + 0.948iT - 73T^{2} \)
79 \( 1 - 3.68T + 79T^{2} \)
83 \( 1 - 8.84iT - 83T^{2} \)
89 \( 1 + 2.17T + 89T^{2} \)
97 \( 1 - 10.0T + 97T^{2} \)
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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

−10.75012355624937602985442765367, −9.784950528485625146279976630645, −9.576405896841661164702530139698, −8.166016900900422013877339159343, −7.52264077931741172795342345342, −6.74901799393198810623574768409, −5.57248639573887012099862973258, −3.51294624832701133913199786393, −1.95658174066757373787104794930, −0.49964211820904333342305411920, 1.98959769683022163589812891916, 2.54322401769738026064812087550, 5.17661162698985753484932390926, 6.28937173282659556940594578052, 7.17685894519685458330011115449, 8.187559293378766475466792866250, 9.122780448259675206535417238191, 9.585748337660479119006296618010, 10.30921109418140723017905745112, 11.47194858508232348990822355777

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