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Properties

Label 2-380-76.31-c1-0-28
Degree $2$
Conductor $380$
Sign $-0.389 + 0.920i$
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  + (−0.621 + 1.27i)2-s + (−0.717 − 1.24i)3-s + (−1.22 − 1.57i)4-s + (−0.5 − 0.866i)5-s + (2.02 − 0.138i)6-s + 2.27i·7-s + (2.76 − 0.577i)8-s + (0.470 − 0.814i)9-s + (1.41 − 0.0968i)10-s + 3.14i·11-s + (−1.08 + 2.65i)12-s + (−3.81 − 2.20i)13-s + (−2.89 − 1.41i)14-s + (−0.717 + 1.24i)15-s + (−0.987 + 3.87i)16-s + (−3.77 − 6.54i)17-s + ⋯
L(s)  = 1  + (−0.439 + 0.898i)2-s + (−0.414 − 0.717i)3-s + (−0.613 − 0.789i)4-s + (−0.223 − 0.387i)5-s + (0.826 − 0.0567i)6-s + 0.860i·7-s + (0.978 − 0.204i)8-s + (0.156 − 0.271i)9-s + (0.446 − 0.0306i)10-s + 0.947i·11-s + (−0.312 + 0.767i)12-s + (−1.05 − 0.610i)13-s + (−0.772 − 0.378i)14-s + (−0.185 + 0.320i)15-s + (−0.246 + 0.969i)16-s + (−0.916 − 1.58i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.389 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.389 + 0.920i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.389 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.211325 - 0.318851i\)
\(L(\frac12)\) \(\approx\) \(0.211325 - 0.318851i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.621 - 1.27i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.366 + 4.34i)T \)
good3 \( 1 + (0.717 + 1.24i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 2.27iT - 7T^{2} \)
11 \( 1 - 3.14iT - 11T^{2} \)
13 \( 1 + (3.81 + 2.20i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.77 + 6.54i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3.92 + 2.26i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.79 + 2.18i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.49T + 31T^{2} \)
37 \( 1 - 7.02iT - 37T^{2} \)
41 \( 1 + (-5.57 + 3.22i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.32 - 4.22i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-7.71 - 4.45i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.50 + 3.17i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.92 + 5.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.70 + 9.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.63 + 6.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.87 + 6.70i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.63 - 11.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.81 - 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.7iT - 83T^{2} \)
89 \( 1 + (-2.24 - 1.29i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.67 - 0.967i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.16554637827610433453532800174, −9.641167971921482053717071408360, −9.330064443548077173899009315218, −8.057655968397269458829861916731, −7.21847462037098645619545368570, −6.53843580097513799550534431607, −5.32849117754144308345666704007, −4.57891614503767583948722642287, −2.22923978604794296372943986305, −0.29997416930677086494213637520, 1.97726972612805340659948259871, 3.78199851105911710519010159389, 4.20887321187548980225160306535, 5.67462459287489910041086291060, 7.19393552578494849534423836546, 8.046880286223590413852244942335, 9.162346765181750414902100359651, 10.20544057466599509409858228818, 10.64020085740413760384780608824, 11.28866715377959457835669970230

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