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Properties

Label 2-387-43.17-c1-0-8
Degree $2$
Conductor $387$
Sign $0.567 - 0.823i$
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  + (0.515 + 2.25i)2-s + (−3.03 + 1.46i)4-s + (1.48 − 3.79i)5-s + (1.38 − 2.40i)7-s + (−1.98 − 2.48i)8-s + (9.33 + 1.40i)10-s + (0.678 + 0.326i)11-s + (1.70 − 0.256i)13-s + (6.15 + 1.89i)14-s + (0.394 − 0.495i)16-s + (1.18 + 3.02i)17-s + (−0.0395 − 0.0269i)19-s + (1.02 + 13.6i)20-s + (−0.388 + 1.70i)22-s + (−0.152 − 2.03i)23-s + ⋯
L(s)  = 1  + (0.364 + 1.59i)2-s + (−1.51 + 0.731i)4-s + (0.665 − 1.69i)5-s + (0.525 − 0.909i)7-s + (−0.701 − 0.880i)8-s + (2.95 + 0.444i)10-s + (0.204 + 0.0984i)11-s + (0.471 − 0.0711i)13-s + (1.64 + 0.507i)14-s + (0.0987 − 0.123i)16-s + (0.288 + 0.733i)17-s + (−0.00906 − 0.00617i)19-s + (0.229 + 3.06i)20-s + (−0.0827 + 0.362i)22-s + (−0.0317 − 0.423i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.567 - 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $0.567 - 0.823i$
Analytic conductor: \(3.09021\)
Root analytic conductor: \(1.75789\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :1/2),\ 0.567 - 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59913 + 0.839630i\)
\(L(\frac12)\) \(\approx\) \(1.59913 + 0.839630i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.39 + 4.86i)T \)
good2 \( 1 + (-0.515 - 2.25i)T + (-1.80 + 0.867i)T^{2} \)
5 \( 1 + (-1.48 + 3.79i)T + (-3.66 - 3.40i)T^{2} \)
7 \( 1 + (-1.38 + 2.40i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.678 - 0.326i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-1.70 + 0.256i)T + (12.4 - 3.83i)T^{2} \)
17 \( 1 + (-1.18 - 3.02i)T + (-12.4 + 11.5i)T^{2} \)
19 \( 1 + (0.0395 + 0.0269i)T + (6.94 + 17.6i)T^{2} \)
23 \( 1 + (0.152 + 2.03i)T + (-22.7 + 3.42i)T^{2} \)
29 \( 1 + (0.714 + 0.220i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (-2.52 + 2.34i)T + (2.31 - 30.9i)T^{2} \)
37 \( 1 + (-3.91 - 6.78i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.86 - 8.18i)T + (-36.9 + 17.7i)T^{2} \)
47 \( 1 + (7.05 - 3.39i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-1.76 - 0.266i)T + (50.6 + 15.6i)T^{2} \)
59 \( 1 + (3.60 - 4.52i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-2.15 - 2.00i)T + (4.55 + 60.8i)T^{2} \)
67 \( 1 + (-4.62 - 3.15i)T + (24.4 + 62.3i)T^{2} \)
71 \( 1 + (-0.543 + 7.25i)T + (-70.2 - 10.5i)T^{2} \)
73 \( 1 + (10.0 - 1.51i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (6.70 - 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.42 - 1.67i)T + (68.5 - 46.7i)T^{2} \)
89 \( 1 + (-11.7 + 3.63i)T + (73.5 - 50.1i)T^{2} \)
97 \( 1 + (-9.41 - 4.53i)T + (60.4 + 75.8i)T^{2} \)
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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

−11.68205642332545052280557811312, −10.28104603641295772662413855627, −9.253609493631268448374870645477, −8.329738470235767639547761216886, −7.892606386514496813411188195638, −6.54948052945730006988103188438, −5.72998185014679587096888746876, −4.76332217767610622829320511584, −4.16416862500100873280233308905, −1.32120810468943354319045892911, 1.86136060797732664606764299263, 2.74553320756985857138374709653, 3.63180252717721533996062308040, 5.16936223909544672063218075053, 6.18717736938307204462538898587, 7.37331321437474545895952813344, 8.889116600116955731751825628782, 9.775316648854065303998519730284, 10.47720545093681269760598407846, 11.37334901069924597689574188120

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