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Properties

Label 2-387-43.36-c1-0-4
Degree $2$
Conductor $387$
Sign $0.0207 - 0.999i$
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  + 0.891·2-s − 1.20·4-s + (1.55 + 2.69i)5-s + (−0.888 + 1.53i)7-s − 2.85·8-s + (1.38 + 2.40i)10-s + 0.636·11-s + (−1.74 + 3.02i)13-s + (−0.792 + 1.37i)14-s − 0.140·16-s + (−0.318 + 0.551i)17-s + (3.31 + 5.74i)19-s + (−1.87 − 3.24i)20-s + 0.567·22-s + (−2.00 − 3.46i)23-s + ⋯
L(s)  = 1  + 0.630·2-s − 0.602·4-s + (0.696 + 1.20i)5-s + (−0.335 + 0.581i)7-s − 1.01·8-s + (0.439 + 0.760i)10-s + 0.191·11-s + (−0.484 + 0.838i)13-s + (−0.211 + 0.366i)14-s − 0.0351·16-s + (−0.0771 + 0.133i)17-s + (0.760 + 1.31i)19-s + (−0.419 − 0.726i)20-s + 0.121·22-s + (−0.417 − 0.723i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0207 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0207 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $0.0207 - 0.999i$
Analytic conductor: \(3.09021\)
Root analytic conductor: \(1.75789\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :1/2),\ 0.0207 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07118 + 1.04919i\)
\(L(\frac12)\) \(\approx\) \(1.07118 + 1.04919i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.83 + 6.29i)T \)
good2 \( 1 - 0.891T + 2T^{2} \)
5 \( 1 + (-1.55 - 2.69i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.888 - 1.53i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 0.636T + 11T^{2} \)
13 \( 1 + (1.74 - 3.02i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.318 - 0.551i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.31 - 5.74i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.00 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.33 + 5.77i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.03 - 1.78i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.92 - 3.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.08T + 41T^{2} \)
47 \( 1 - 1.07T + 47T^{2} \)
53 \( 1 + (2.66 + 4.62i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 6.17T + 59T^{2} \)
61 \( 1 + (3.33 - 5.78i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.87 + 10.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.64 + 8.04i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.03 + 1.78i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.65 + 9.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.52 + 13.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.50 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.91T + 97T^{2} \)
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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

−11.86802637793805990784933763520, −10.50341685582313244306507006241, −9.760252831609581382695114719612, −9.031944586156496212014878289555, −7.74586870950973143630868043104, −6.35809829240360734328130676692, −5.98598270034883058125235445057, −4.64025469973654180637484484519, −3.42555184386889721341103620143, −2.33776926105175073260305037307, 0.863108613896391573526491203364, 2.98197287090303814967443341998, 4.35052264910976377203055773427, 5.13383462474481833217323863428, 5.91457674602109950522342131523, 7.30479212279395090655007583528, 8.521185533931940773686504593170, 9.392110604952596304178196656717, 9.868463164202624301735554322231, 11.23633094525538487595386265598

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