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Properties

Label 2-380-76.27-c1-0-4
Degree $2$
Conductor $380$
Sign $-0.643 - 0.765i$
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  + (−1.31 − 0.518i)2-s + (−0.736 + 1.27i)3-s + (1.46 + 1.36i)4-s + (−0.5 + 0.866i)5-s + (1.63 − 1.29i)6-s − 0.456i·7-s + (−1.21 − 2.55i)8-s + (0.414 + 0.717i)9-s + (1.10 − 0.880i)10-s + 4.12i·11-s + (−2.81 + 0.861i)12-s + (1.19 − 0.690i)13-s + (−0.236 + 0.600i)14-s + (−0.736 − 1.27i)15-s + (0.279 + 3.99i)16-s + (2.62 − 4.54i)17-s + ⋯
L(s)  = 1  + (−0.930 − 0.366i)2-s + (−0.425 + 0.736i)3-s + (0.731 + 0.681i)4-s + (−0.223 + 0.387i)5-s + (0.665 − 0.529i)6-s − 0.172i·7-s + (−0.430 − 0.902i)8-s + (0.138 + 0.239i)9-s + (0.349 − 0.278i)10-s + 1.24i·11-s + (−0.813 + 0.248i)12-s + (0.331 − 0.191i)13-s + (−0.0631 + 0.160i)14-s + (−0.190 − 0.329i)15-s + (0.0698 + 0.997i)16-s + (0.636 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.643 - 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.643 - 0.765i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.643 - 0.765i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.221307 + 0.475454i\)
\(L(\frac12)\) \(\approx\) \(0.221307 + 0.475454i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 + 0.518i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (4.15 - 1.30i)T \)
good3 \( 1 + (0.736 - 1.27i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 0.456iT - 7T^{2} \)
11 \( 1 - 4.12iT - 11T^{2} \)
13 \( 1 + (-1.19 + 0.690i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.62 + 4.54i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (7.10 - 4.10i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (8.58 - 4.95i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.72T + 31T^{2} \)
37 \( 1 - 3.87iT - 37T^{2} \)
41 \( 1 + (0.344 + 0.198i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.80 - 5.08i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.49 - 4.32i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.85 - 1.64i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.81 + 3.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.29 + 3.97i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.68 + 9.84i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.118 + 0.204i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.15 - 10.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.38 + 12.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.73iT - 83T^{2} \)
89 \( 1 + (-7.61 + 4.39i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.52 - 0.877i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.39939822039567979760719453334, −10.61891835749577837202533016573, −9.934500420566536125798738905244, −9.319670968367431117229493863577, −7.85241467243961386085535483875, −7.34847293037338014605588542049, −6.04555262358161682240639810794, −4.61467308020497409049353522017, −3.52966620216208197040673376963, −1.94474788926731246831729240016, 0.48734153759159565537439124556, 1.95634861310335454370093548765, 3.92396139056868834985261678340, 5.93235645033870763503577776721, 6.06397923515166404752712136875, 7.38575123434719808476277717475, 8.277002722274169586133546297869, 8.909279086690119019523827716445, 10.09889914897061123439389869497, 11.01434443829004280161018994115

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