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Properties

Label 2-387-43.6-c1-0-1
Degree $2$
Conductor $387$
Sign $0.343 - 0.938i$
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-s − 4-s + (−0.5 + 0.866i)5-s + (−1.5 − 2.59i)7-s + 3·8-s + (0.5 − 0.866i)10-s + (2.5 + 4.33i)13-s + (1.5 + 2.59i)14-s − 16-s + (1.5 + 2.59i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)20-s + (−3.5 + 6.06i)23-s + (2 + 3.46i)25-s + (−2.5 − 4.33i)26-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.5·4-s + (−0.223 + 0.387i)5-s + (−0.566 − 0.981i)7-s + 1.06·8-s + (0.158 − 0.273i)10-s + (0.693 + 1.20i)13-s + (0.400 + 0.694i)14-s − 0.250·16-s + (0.363 + 0.630i)17-s + (−0.114 + 0.198i)19-s + (0.111 − 0.193i)20-s + (−0.729 + 1.26i)23-s + (0.400 + 0.692i)25-s + (−0.490 − 0.849i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.343 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $0.343 - 0.938i$
Analytic conductor: \(3.09021\)
Root analytic conductor: \(1.75789\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :1/2),\ 0.343 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.517683 + 0.361686i\)
\(L(\frac12)\) \(\approx\) \(0.517683 + 0.361686i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4 - 5.19i)T \)
good2 \( 1 + T + 2T^{2} \)
5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.5 - 6.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.5 + 7.79i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + (2.5 - 4.33i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−11.09462155300758319897520107671, −10.57056392768403949607736870881, −9.573386552916146166544281910704, −8.938664893431906517516752534196, −7.73863488900462569984602735480, −7.10391791558200801367221049154, −5.89669917239444553617782279913, −4.30385207335211798804657401971, −3.58071079634792758129270294535, −1.38250738651934570030117821060, 0.61088747845169900649316933416, 2.68155491891568906174088931547, 4.18353268518270748338903154688, 5.34459878062701987007055727537, 6.34740161396081794023446720278, 7.87025459757384297811958532957, 8.416338582271964854707702557315, 9.264506488341810633380950090586, 10.03275525722517290276607768072, 10.93265015643147667652644050703

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