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Properties

Label 2-387-43.2-c2-0-10
Degree $2$
Conductor $387$
Sign $0.858 - 0.512i$
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  + (−0.311 − 0.647i)2-s + (2.17 − 2.72i)4-s + (3.49 + 0.798i)5-s + 9.27i·7-s + (−5.24 − 1.19i)8-s + (−0.573 − 2.51i)10-s + (7.85 + 9.85i)11-s + (−5.05 + 22.1i)13-s + (6.00 − 2.89i)14-s + (−2.23 − 9.81i)16-s + (4.36 + 19.1i)17-s + (2.63 + 2.09i)19-s + (9.77 − 7.79i)20-s + (3.93 − 8.16i)22-s + (−9.12 − 11.4i)23-s + ⋯
L(s)  = 1  + (−0.155 − 0.323i)2-s + (0.542 − 0.680i)4-s + (0.699 + 0.159i)5-s + 1.32i·7-s + (−0.655 − 0.149i)8-s + (−0.0573 − 0.251i)10-s + (0.714 + 0.895i)11-s + (−0.388 + 1.70i)13-s + (0.429 − 0.206i)14-s + (−0.139 − 0.613i)16-s + (0.256 + 1.12i)17-s + (0.138 + 0.110i)19-s + (0.488 − 0.389i)20-s + (0.178 − 0.371i)22-s + (−0.396 − 0.497i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $0.858 - 0.512i$
Analytic conductor: \(10.5449\)
Root analytic conductor: \(3.24730\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :1),\ 0.858 - 0.512i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.81121 + 0.499272i\)
\(L(\frac12)\) \(\approx\) \(1.81121 + 0.499272i\)
\(L(2)\) 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 \( 1 + (-42.3 + 7.19i)T \)
good2 \( 1 + (0.311 + 0.647i)T + (-2.49 + 3.12i)T^{2} \)
5 \( 1 + (-3.49 - 0.798i)T + (22.5 + 10.8i)T^{2} \)
7 \( 1 - 9.27iT - 49T^{2} \)
11 \( 1 + (-7.85 - 9.85i)T + (-26.9 + 117. i)T^{2} \)
13 \( 1 + (5.05 - 22.1i)T + (-152. - 73.3i)T^{2} \)
17 \( 1 + (-4.36 - 19.1i)T + (-260. + 125. i)T^{2} \)
19 \( 1 + (-2.63 - 2.09i)T + (80.3 + 351. i)T^{2} \)
23 \( 1 + (9.12 + 11.4i)T + (-117. + 515. i)T^{2} \)
29 \( 1 + (0.220 + 0.457i)T + (-524. + 657. i)T^{2} \)
31 \( 1 + (6.91 - 3.33i)T + (599. - 751. i)T^{2} \)
37 \( 1 + 40.3iT - 1.36e3T^{2} \)
41 \( 1 + (-44.6 + 21.5i)T + (1.04e3 - 1.31e3i)T^{2} \)
47 \( 1 + (-21.6 + 27.1i)T + (-491. - 2.15e3i)T^{2} \)
53 \( 1 + (-16.3 - 71.4i)T + (-2.53e3 + 1.21e3i)T^{2} \)
59 \( 1 + (1.19 + 5.22i)T + (-3.13e3 + 1.51e3i)T^{2} \)
61 \( 1 + (-34.7 + 72.0i)T + (-2.32e3 - 2.90e3i)T^{2} \)
67 \( 1 + (37.1 - 46.5i)T + (-998. - 4.37e3i)T^{2} \)
71 \( 1 + (-4.50 - 3.58i)T + (1.12e3 + 4.91e3i)T^{2} \)
73 \( 1 + (-22.3 - 5.09i)T + (4.80e3 + 2.31e3i)T^{2} \)
79 \( 1 - 77.7T + 6.24e3T^{2} \)
83 \( 1 + (106. + 51.3i)T + (4.29e3 + 5.38e3i)T^{2} \)
89 \( 1 + (-42.5 + 88.2i)T + (-4.93e3 - 6.19e3i)T^{2} \)
97 \( 1 + (13.0 + 16.3i)T + (-2.09e3 + 9.17e3i)T^{2} \)
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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

−11.21037298706601489369576261133, −10.17116036685040941121553464023, −9.398985682889338073534871070138, −8.891296605304695299745616539749, −7.21950951946226002085499507890, −6.24005242658840830784104223134, −5.67901116070751874766082237884, −4.21424935602110570792846530151, −2.29622861040179200801378938171, −1.83025658969354365780170842777, 0.881287757819117820756669318366, 2.81214504622559337854620501217, 3.81503593983631202584514580751, 5.39694187288274289463405444406, 6.35013719531386264495225432183, 7.43620194573112110808330931696, 7.957010617287216378689983674863, 9.210736725862999934665102993979, 10.10101445786655176706015409458, 11.02855798540632962104532731868

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