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Properties

Label 2-387-43.37-c2-0-24
Degree $2$
Conductor $387$
Sign $0.837 + 0.546i$
Analytic cond. $10.5449$
Root an. cond. $3.24730$
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  + 0.780i·2-s + 3.39·4-s + (−4.98 − 2.87i)5-s + (7.65 − 4.41i)7-s + 5.77i·8-s + (2.24 − 3.88i)10-s + 1.24·11-s + (−4.72 − 8.18i)13-s + (3.45 + 5.97i)14-s + 9.05·16-s + (−3.12 − 5.41i)17-s + (−1.67 − 0.968i)19-s + (−16.8 − 9.74i)20-s + 0.968i·22-s + (17.7 − 30.8i)23-s + ⋯
L(s)  = 1  + 0.390i·2-s + 0.847·4-s + (−0.996 − 0.575i)5-s + (1.09 − 0.631i)7-s + 0.721i·8-s + (0.224 − 0.388i)10-s + 0.112·11-s + (−0.363 − 0.629i)13-s + (0.246 + 0.426i)14-s + 0.565·16-s + (−0.183 − 0.318i)17-s + (−0.0883 − 0.0509i)19-s + (−0.844 − 0.487i)20-s + 0.0440i·22-s + (0.773 − 1.33i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $0.837 + 0.546i$
Analytic conductor: \(10.5449\)
Root analytic conductor: \(3.24730\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :1),\ 0.837 + 0.546i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.82952 - 0.544041i\)
\(L(\frac12)\) \(\approx\) \(1.82952 - 0.544041i\)
\(L(2)\) 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 \( 1 + (-42.9 + 1.36i)T \)
good2 \( 1 - 0.780iT - 4T^{2} \)
5 \( 1 + (4.98 + 2.87i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-7.65 + 4.41i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 - 1.24T + 121T^{2} \)
13 \( 1 + (4.72 + 8.18i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (3.12 + 5.41i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (1.67 + 0.968i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-17.7 + 30.8i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-25.0 + 14.4i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-6.52 + 11.3i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-4.60 - 2.66i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 25.8T + 1.68e3T^{2} \)
47 \( 1 - 73.9T + 2.20e3T^{2} \)
53 \( 1 + (35.2 - 61.0i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + 26.1T + 3.48e3T^{2} \)
61 \( 1 + (51.7 - 29.9i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-17.2 + 29.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-37.5 + 21.7i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-17.0 + 9.85i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-19.0 - 33.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-46.1 + 79.9i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (27.7 + 16.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 112.T + 9.40e3T^{2} \)
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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

−11.01491925639492232184620323182, −10.46710830394022150667688446583, −8.857097764878891112392679056435, −7.930669071828630217279968763782, −7.53852634032440795734453575825, −6.40214666665310465768977595812, −5.03754479009939719256890259471, −4.25083036503317066894600997687, −2.62568110265486234806713792658, −0.905263540693455013883941840107, 1.58864060427185686784554949834, 2.84248517840872379887805791127, 4.00220177413623111003119063604, 5.30437700595027719797716635536, 6.66474838984737766194865804099, 7.43576696345233648202447512503, 8.272173804983233702838671339140, 9.428319564166276707333720119578, 10.73158248190146644279299081032, 11.23422863755443870429454213514

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