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Properties

Label 2-387-43.39-c2-0-27
Degree $2$
Conductor $387$
Sign $0.995 + 0.0926i$
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  + (2.20 + 1.75i)2-s + (0.878 + 3.84i)4-s + (1.34 − 2.79i)5-s − 7.15i·7-s + (0.0627 − 0.130i)8-s + (7.88 − 3.79i)10-s + (3.85 − 16.8i)11-s + (6.32 + 3.04i)13-s + (12.5 − 15.7i)14-s + (14.5 − 7.02i)16-s + (−23.3 + 11.2i)17-s + (−4.11 + 0.939i)19-s + (11.9 + 2.72i)20-s + (38.1 − 30.4i)22-s + (4.20 − 18.4i)23-s + ⋯
L(s)  = 1  + (1.10 + 0.878i)2-s + (0.219 + 0.962i)4-s + (0.269 − 0.559i)5-s − 1.02i·7-s + (0.00783 − 0.0162i)8-s + (0.788 − 0.379i)10-s + (0.350 − 1.53i)11-s + (0.486 + 0.234i)13-s + (0.898 − 1.12i)14-s + (0.912 − 0.439i)16-s + (−1.37 + 0.660i)17-s + (−0.216 + 0.0494i)19-s + (0.597 + 0.136i)20-s + (1.73 − 1.38i)22-s + (0.182 − 0.801i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0926i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $0.995 + 0.0926i$
Analytic conductor: \(10.5449\)
Root analytic conductor: \(3.24730\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :1),\ 0.995 + 0.0926i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.15148 - 0.146229i\)
\(L(\frac12)\) \(\approx\) \(3.15148 - 0.146229i\)
\(L(2)\) 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 \( 1 + (42.9 - 1.09i)T \)
good2 \( 1 + (-2.20 - 1.75i)T + (0.890 + 3.89i)T^{2} \)
5 \( 1 + (-1.34 + 2.79i)T + (-15.5 - 19.5i)T^{2} \)
7 \( 1 + 7.15iT - 49T^{2} \)
11 \( 1 + (-3.85 + 16.8i)T + (-109. - 52.4i)T^{2} \)
13 \( 1 + (-6.32 - 3.04i)T + (105. + 132. i)T^{2} \)
17 \( 1 + (23.3 - 11.2i)T + (180. - 225. i)T^{2} \)
19 \( 1 + (4.11 - 0.939i)T + (325. - 156. i)T^{2} \)
23 \( 1 + (-4.20 + 18.4i)T + (-476. - 229. i)T^{2} \)
29 \( 1 + (-35.1 - 28.0i)T + (187. + 819. i)T^{2} \)
31 \( 1 + (14.3 - 17.9i)T + (-213. - 936. i)T^{2} \)
37 \( 1 - 21.1iT - 1.36e3T^{2} \)
41 \( 1 + (6.03 - 7.56i)T + (-374. - 1.63e3i)T^{2} \)
47 \( 1 + (-2.02 - 8.85i)T + (-1.99e3 + 958. i)T^{2} \)
53 \( 1 + (-50.5 + 24.3i)T + (1.75e3 - 2.19e3i)T^{2} \)
59 \( 1 + (-6.59 + 3.17i)T + (2.17e3 - 2.72e3i)T^{2} \)
61 \( 1 + (22.4 - 17.8i)T + (828. - 3.62e3i)T^{2} \)
67 \( 1 + (-1.18 - 5.18i)T + (-4.04e3 + 1.94e3i)T^{2} \)
71 \( 1 + (-36.9 + 8.42i)T + (4.54e3 - 2.18e3i)T^{2} \)
73 \( 1 + (-41.8 + 86.8i)T + (-3.32e3 - 4.16e3i)T^{2} \)
79 \( 1 + 33.9T + 6.24e3T^{2} \)
83 \( 1 + (-53.2 - 66.8i)T + (-1.53e3 + 6.71e3i)T^{2} \)
89 \( 1 + (87.7 - 69.9i)T + (1.76e3 - 7.72e3i)T^{2} \)
97 \( 1 + (3.30 - 14.4i)T + (-8.47e3 - 4.08e3i)T^{2} \)
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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

−11.07481590446034109253261817045, −10.42058776991175839429178564103, −8.914136697213296072116118779661, −8.275782837473455719866162083024, −6.81208905023276006084830103952, −6.40296964525896273162958022791, −5.22362021700636768305857254723, −4.30455138185327047686479129532, −3.38695937967122102388896855171, −1.07047973665648151542199364394, 2.01966519108628181717533069601, 2.69573710750072333716204067206, 4.10644005287525648235413309898, 4.99882387814264801512264550042, 6.08909386389929064213381632462, 7.09158530184441193434088483980, 8.508514378986172512714690646864, 9.558982983980885181887312995004, 10.46830554393383810511295835590, 11.42838830582698877722054959881

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