Properties

Label 2-380-95.33-c1-0-9
Degree $2$
Conductor $380$
Sign $-0.860 - 0.509i$
Analytic cond. $3.03431$
Root an. cond. $1.74192$
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  + (−1.41 − 2.02i)3-s + (0.187 − 2.22i)5-s + (−3.90 − 1.04i)7-s + (−1.06 + 2.93i)9-s + (−1.50 + 2.60i)11-s + (2.49 + 1.74i)13-s + (−4.78 + 2.78i)15-s + (1.54 − 3.31i)17-s + (−2.24 + 3.73i)19-s + (3.42 + 9.40i)21-s + (4.38 + 0.383i)23-s + (−4.92 − 0.837i)25-s + (0.299 − 0.0802i)27-s + (−6.16 − 2.24i)29-s + (0.0797 − 0.0460i)31-s + ⋯
L(s)  = 1  + (−0.819 − 1.17i)3-s + (0.0840 − 0.996i)5-s + (−1.47 − 0.395i)7-s + (−0.356 + 0.978i)9-s + (−0.453 + 0.785i)11-s + (0.692 + 0.484i)13-s + (−1.23 + 0.718i)15-s + (0.374 − 0.803i)17-s + (−0.514 + 0.857i)19-s + (0.747 + 2.05i)21-s + (0.914 + 0.0800i)23-s + (−0.985 − 0.167i)25-s + (0.0576 − 0.0154i)27-s + (−1.14 − 0.416i)29-s + (0.0143 − 0.00826i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 - 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.860 - 0.509i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.860 - 0.509i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.102221 + 0.373443i\)
\(L(\frac12)\) \(\approx\) \(0.102221 + 0.373443i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.187 + 2.22i)T \)
19 \( 1 + (2.24 - 3.73i)T \)
good3 \( 1 + (1.41 + 2.02i)T + (-1.02 + 2.81i)T^{2} \)
7 \( 1 + (3.90 + 1.04i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.50 - 2.60i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.49 - 1.74i)T + (4.44 + 12.2i)T^{2} \)
17 \( 1 + (-1.54 + 3.31i)T + (-10.9 - 13.0i)T^{2} \)
23 \( 1 + (-4.38 - 0.383i)T + (22.6 + 3.99i)T^{2} \)
29 \( 1 + (6.16 + 2.24i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-0.0797 + 0.0460i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.63 + 5.63i)T + 37iT^{2} \)
41 \( 1 + (0.252 + 0.0444i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.986 + 11.2i)T + (-42.3 + 7.46i)T^{2} \)
47 \( 1 + (12.1 - 5.66i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (0.0341 - 0.390i)T + (-52.1 - 9.20i)T^{2} \)
59 \( 1 + (-6.11 + 2.22i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (5.92 - 4.97i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (3.18 + 6.83i)T + (-43.0 + 51.3i)T^{2} \)
71 \( 1 + (-10.3 + 12.3i)T + (-12.3 - 69.9i)T^{2} \)
73 \( 1 + (-4.03 + 2.82i)T + (24.9 - 68.5i)T^{2} \)
79 \( 1 + (-1.67 + 9.48i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (1.31 - 4.91i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-1.38 - 7.83i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (4.48 + 2.09i)T + (62.3 + 74.3i)T^{2} \)
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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

−10.96696636155394358599292293984, −9.835992046408255631364822226520, −9.077096189627270866436217391981, −7.73848632247320479394651701760, −6.93306136531960123429998144640, −6.11286478852584354144182861846, −5.17044536176612897067818440409, −3.69840725654141826691423643343, −1.78497102088699609550114094232, −0.27644158685100807353441088343, 3.01739928380493420931006188868, 3.67160957184058406419239655594, 5.26706898393707482941767180574, 6.08788882218699588069733608886, 6.77698345035942383541900415711, 8.375807041393787289069923549757, 9.507092698580440848808125135802, 10.18462385099119816879735120237, 10.90683242069873663445181409793, 11.42810176068767610512486142953

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