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Properties

Label 2-387-43.15-c1-0-6
Degree $2$
Conductor $387$
Sign $0.0193 - 0.999i$
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  + (0.0594 − 0.0745i)2-s + (0.443 + 1.94i)4-s + (0.284 + 3.80i)5-s + (1.30 − 2.26i)7-s + (0.342 + 0.165i)8-s + (0.300 + 0.204i)10-s + (0.456 − 1.99i)11-s + (−1.76 + 1.20i)13-s + (−0.0912 − 0.232i)14-s + (−3.55 + 1.71i)16-s + (−0.388 + 5.18i)17-s + (−0.603 + 0.559i)19-s + (−7.25 + 2.23i)20-s + (−0.121 − 0.152i)22-s + (2.60 − 0.805i)23-s + ⋯
L(s)  = 1  + (0.0420 − 0.0527i)2-s + (0.221 + 0.970i)4-s + (0.127 + 1.70i)5-s + (0.495 − 0.857i)7-s + (0.121 + 0.0583i)8-s + (0.0950 + 0.0647i)10-s + (0.137 − 0.602i)11-s + (−0.490 + 0.334i)13-s + (−0.0243 − 0.0621i)14-s + (−0.888 + 0.427i)16-s + (−0.0942 + 1.25i)17-s + (−0.138 + 0.128i)19-s + (−1.62 + 0.500i)20-s + (−0.0260 − 0.0326i)22-s + (0.544 − 0.167i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0193 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0193 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $0.0193 - 0.999i$
Analytic conductor: \(3.09021\)
Root analytic conductor: \(1.75789\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :1/2),\ 0.0193 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05716 + 1.03687i\)
\(L(\frac12)\) \(\approx\) \(1.05716 + 1.03687i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.37 + 5.62i)T \)
good2 \( 1 + (-0.0594 + 0.0745i)T + (-0.445 - 1.94i)T^{2} \)
5 \( 1 + (-0.284 - 3.80i)T + (-4.94 + 0.745i)T^{2} \)
7 \( 1 + (-1.30 + 2.26i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.456 + 1.99i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (1.76 - 1.20i)T + (4.74 - 12.1i)T^{2} \)
17 \( 1 + (0.388 - 5.18i)T + (-16.8 - 2.53i)T^{2} \)
19 \( 1 + (0.603 - 0.559i)T + (1.41 - 18.9i)T^{2} \)
23 \( 1 + (-2.60 + 0.805i)T + (19.0 - 12.9i)T^{2} \)
29 \( 1 + (1.43 + 3.65i)T + (-21.2 + 19.7i)T^{2} \)
31 \( 1 + (-7.40 - 1.11i)T + (29.6 + 9.13i)T^{2} \)
37 \( 1 + (0.431 + 0.746i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.40 + 3.02i)T + (-9.12 - 39.9i)T^{2} \)
47 \( 1 + (-0.00195 - 0.00857i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (3.12 + 2.13i)T + (19.3 + 49.3i)T^{2} \)
59 \( 1 + (-3.54 + 1.70i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-6.20 + 0.935i)T + (58.2 - 17.9i)T^{2} \)
67 \( 1 + (9.35 - 8.67i)T + (5.00 - 66.8i)T^{2} \)
71 \( 1 + (-0.220 - 0.0680i)T + (58.6 + 39.9i)T^{2} \)
73 \( 1 + (-9.06 + 6.18i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (0.133 - 0.231i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.09 + 7.88i)T + (-60.8 - 56.4i)T^{2} \)
89 \( 1 + (5.18 - 13.2i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + (-1.14 + 5.00i)T + (-87.3 - 42.0i)T^{2} \)
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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

−11.31213597692583178056528758205, −10.82644481127590424254529861693, −10.03926551086211007323635327699, −8.568372098003513063210208896362, −7.65024783170550745741078345826, −6.96166093937470598011942612722, −6.13634564213307344473725824490, −4.26537857769393277509634889688, −3.40341497237102412477520102139, −2.27321189455954705252345520428, 1.04263966701946857398866251092, 2.35261235712075248108937155057, 4.78704902237750127034705391413, 4.97676180339285968012016462918, 6.03485257468263977336343750700, 7.38545031026124059126673966442, 8.562921208032133328641711596308, 9.314511667318257696635793381276, 9.903212919392061378802019483234, 11.27949852216498799265866214529

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