Properties

Label 2-380-19.16-c1-0-0
Degree $2$
Conductor $380$
Sign $0.399 - 0.916i$
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  + (−1.57 + 0.573i)3-s + (0.173 − 0.984i)5-s + (0.230 + 0.399i)7-s + (−0.142 + 0.119i)9-s + (−0.642 + 1.11i)11-s + (5.58 + 2.03i)13-s + (0.291 + 1.65i)15-s + (4.26 + 3.57i)17-s + (−3.93 + 1.88i)19-s + (−0.592 − 0.497i)21-s + (0.379 + 2.15i)23-s + (−0.939 − 0.342i)25-s + (2.67 − 4.62i)27-s + (−7.31 + 6.14i)29-s + (4.61 + 8.00i)31-s + ⋯
L(s)  = 1  + (−0.910 + 0.331i)3-s + (0.0776 − 0.440i)5-s + (0.0871 + 0.150i)7-s + (−0.0475 + 0.0399i)9-s + (−0.193 + 0.335i)11-s + (1.54 + 0.564i)13-s + (0.0752 + 0.426i)15-s + (1.03 + 0.867i)17-s + (−0.901 + 0.432i)19-s + (−0.129 − 0.108i)21-s + (0.0792 + 0.449i)23-s + (−0.187 − 0.0684i)25-s + (0.514 − 0.890i)27-s + (−1.35 + 1.14i)29-s + (0.829 + 1.43i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.399 - 0.916i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.399 - 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.798593 + 0.523376i\)
\(L(\frac12)\) \(\approx\) \(0.798593 + 0.523376i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (3.93 - 1.88i)T \)
good3 \( 1 + (1.57 - 0.573i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (-0.230 - 0.399i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.642 - 1.11i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.58 - 2.03i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-4.26 - 3.57i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.379 - 2.15i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (7.31 - 6.14i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-4.61 - 8.00i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.90T + 37T^{2} \)
41 \( 1 + (-4.53 + 1.64i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.655 - 3.71i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (5.24 - 4.40i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (1.37 + 7.78i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (4.10 + 3.44i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.76 + 9.98i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-5.05 + 4.24i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-1.19 + 6.78i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (12.3 - 4.47i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (1.30 - 0.474i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (0.0660 + 0.114i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.89 - 1.05i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-0.447 - 0.375i)T + (16.8 + 95.5i)T^{2} \)
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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

−11.32237714921758517361042719170, −10.81267559550351712096907251454, −9.847930689665609341234062788131, −8.727148510132278315689486253534, −7.957570855465190383204470258122, −6.43173796296065897426045853893, −5.74664555817116916322746518753, −4.76553958477620789278035631177, −3.59873599968284434904784867995, −1.54881373653986461206402587647, 0.78806876298540525708994465675, 2.83059485781771187390290782161, 4.20462601640062894757055954378, 5.82797088063163681080996520787, 6.04953438169827061999093911412, 7.34056893841116461669672036445, 8.282535985081280516390377703754, 9.448995785539181532552743861227, 10.58361024025011035142788050351, 11.22755248585852730496827252601

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