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Properties

Label 2-380-95.49-c1-0-6
Degree $2$
Conductor $380$
Sign $-0.775 + 0.631i$
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  + (−2.48 − 1.43i)3-s + (1.38 + 1.75i)5-s − 3.54i·7-s + (2.62 + 4.54i)9-s − 1.81·11-s + (2.78 − 1.60i)13-s + (−0.934 − 6.35i)15-s + (−6.92 − 3.99i)17-s + (0.863 − 4.27i)19-s + (−5.09 + 8.82i)21-s + (−7.30 + 4.21i)23-s + (−1.14 + 4.86i)25-s − 6.46i·27-s + (−4.29 − 7.43i)29-s − 1.70·31-s + ⋯
L(s)  = 1  + (−1.43 − 0.829i)3-s + (0.620 + 0.784i)5-s − 1.34i·7-s + (0.875 + 1.51i)9-s − 0.547·11-s + (0.771 − 0.445i)13-s + (−0.241 − 1.64i)15-s + (−1.67 − 0.969i)17-s + (0.198 − 0.980i)19-s + (−1.11 + 1.92i)21-s + (−1.52 + 0.878i)23-s + (−0.229 + 0.973i)25-s − 1.24i·27-s + (−0.796 − 1.38i)29-s − 0.306·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.775 + 0.631i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.775 + 0.631i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.197385 - 0.555425i\)
\(L(\frac12)\) \(\approx\) \(0.197385 - 0.555425i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.38 - 1.75i)T \)
19 \( 1 + (-0.863 + 4.27i)T \)
good3 \( 1 + (2.48 + 1.43i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + 3.54iT - 7T^{2} \)
11 \( 1 + 1.81T + 11T^{2} \)
13 \( 1 + (-2.78 + 1.60i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (6.92 + 3.99i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (7.30 - 4.21i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.29 + 7.43i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.70T + 31T^{2} \)
37 \( 1 + 5.50iT - 37T^{2} \)
41 \( 1 + (-4.05 + 7.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.35 - 2.51i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.16 + 0.674i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.92 - 1.10i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.960 + 1.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.83 - 4.90i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.04 + 4.64i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.94 - 5.09i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.82 + 1.63i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.08 + 3.61i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.30iT - 83T^{2} \)
89 \( 1 + (-2.73 - 4.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.91 - 3.99i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.03722233607678918668325280961, −10.51897708577945155321286798029, −9.431915798212045909222888672698, −7.63857790969589912406501618673, −7.11698474663632002739729203687, −6.24966509590122256499212404953, −5.44573027839190507041563643093, −4.08418506058393439677131345912, −2.18720854172912261027715238931, −0.45237086967773660570062705820, 1.94259351280237042902666351107, 4.09302356425324325545397966114, 5.03104065290484363279162753819, 5.94801116657910300815463333597, 6.28516514858493415067293846072, 8.370043778414759129644435158363, 9.050568704729763441761427750557, 10.02189951175850464685755169300, 10.81866278360798483127604743173, 11.64720215031643004593738449228

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