Properties

Label 2-380-19.9-c1-0-3
Degree $2$
Conductor $380$
Sign $-0.343 + 0.939i$
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  + (−2.10 + 1.76i)3-s + (−0.939 − 0.342i)5-s + (−1.23 + 2.14i)7-s + (0.790 − 4.48i)9-s + (0.186 + 0.322i)11-s + (−2.64 − 2.21i)13-s + (2.58 − 0.940i)15-s + (−0.361 − 2.05i)17-s + (0.0754 − 4.35i)19-s + (−1.18 − 6.70i)21-s + (−0.815 + 0.296i)23-s + (0.766 + 0.642i)25-s + (2.13 + 3.70i)27-s + (1.80 − 10.2i)29-s + (−0.935 + 1.62i)31-s + ⋯
L(s)  = 1  + (−1.21 + 1.02i)3-s + (−0.420 − 0.152i)5-s + (−0.467 + 0.810i)7-s + (0.263 − 1.49i)9-s + (0.0562 + 0.0973i)11-s + (−0.732 − 0.614i)13-s + (0.666 − 0.242i)15-s + (−0.0877 − 0.497i)17-s + (0.0173 − 0.999i)19-s + (−0.257 − 1.46i)21-s + (−0.170 + 0.0618i)23-s + (0.153 + 0.128i)25-s + (0.411 + 0.712i)27-s + (0.334 − 1.89i)29-s + (−0.168 + 0.291i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.343 + 0.939i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.343 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0674304 - 0.0964561i\)
\(L(\frac12)\) \(\approx\) \(0.0674304 - 0.0964561i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (-0.0754 + 4.35i)T \)
good3 \( 1 + (2.10 - 1.76i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (1.23 - 2.14i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.186 - 0.322i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.64 + 2.21i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.361 + 2.05i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (0.815 - 0.296i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.80 + 10.2i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (0.935 - 1.62i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.21T + 37T^{2} \)
41 \( 1 + (6.08 - 5.10i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (3.27 + 1.19i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.871 - 4.94i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-7.54 + 2.74i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.44 + 8.18i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (12.1 - 4.42i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.28 - 7.31i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (2.31 + 0.844i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (3.87 - 3.25i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (11.9 - 9.99i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (3.14 - 5.45i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.811 + 0.680i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (2.49 + 14.1i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.19169136303932865733077832233, −10.10996234607237058175458418791, −9.564795362444431450771634334394, −8.475955394345835371370776595724, −7.10473089493297885404679335188, −6.00139398000516539654687721414, −5.15661000643997072519573257096, −4.34302971963499136339591761650, −2.88398827360393141804576004386, −0.090784322932838528747600465566, 1.60302020002330042788057809140, 3.60260701654841659801020816990, 4.94075204215777307160653495272, 6.08529514881699287580334887424, 6.97663433996894240719918894879, 7.43810163236729405529769045596, 8.740514353979879820067374141258, 10.24230584052478991162112822270, 10.73218562282358994688532212496, 11.93269448338930411646374803688

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