Properties

Label 2-380-19.4-c1-0-1
Degree $2$
Conductor $380$
Sign $0.173 - 0.984i$
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  + (0.166 + 0.942i)3-s + (0.766 − 0.642i)5-s + (−2.28 + 3.95i)7-s + (1.95 − 0.712i)9-s + (−0.558 − 0.968i)11-s + (−0.897 + 5.08i)13-s + (0.733 + 0.615i)15-s + (6.80 + 2.47i)17-s + (−3.68 + 2.32i)19-s + (−4.10 − 1.49i)21-s + (−3.31 − 2.78i)23-s + (0.173 − 0.984i)25-s + (2.43 + 4.21i)27-s + (2.51 − 0.913i)29-s + (1.37 − 2.37i)31-s + ⋯
L(s)  = 1  + (0.0959 + 0.544i)3-s + (0.342 − 0.287i)5-s + (−0.862 + 1.49i)7-s + (0.652 − 0.237i)9-s + (−0.168 − 0.291i)11-s + (−0.248 + 1.41i)13-s + (0.189 + 0.158i)15-s + (1.64 + 0.600i)17-s + (−0.845 + 0.533i)19-s + (−0.895 − 0.325i)21-s + (−0.692 − 0.580i)23-s + (0.0347 − 0.196i)25-s + (0.468 + 0.810i)27-s + (0.466 − 0.169i)29-s + (0.246 − 0.426i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03427 + 0.867747i\)
\(L(\frac12)\) \(\approx\) \(1.03427 + 0.867747i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (3.68 - 2.32i)T \)
good3 \( 1 + (-0.166 - 0.942i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (2.28 - 3.95i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.558 + 0.968i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.897 - 5.08i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-6.80 - 2.47i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (3.31 + 2.78i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-2.51 + 0.913i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-1.37 + 2.37i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.491T + 37T^{2} \)
41 \( 1 + (-1.45 - 8.24i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-3.16 + 2.65i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-2.48 + 0.904i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (10.4 + 8.76i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-2.45 - 0.892i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-3.64 - 3.06i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-11.9 + 4.33i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-0.524 + 0.439i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (2.28 + 12.9i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (2.75 + 15.6i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-1.88 + 3.25i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.04 + 11.5i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-3.27 - 1.19i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.79646095315281768416555174791, −10.31333102717858020802935441548, −9.697507777119690418208865536160, −9.019799322077524048482038317831, −8.081056968919908048991417103137, −6.50029181965070014252621449764, −5.88372976576412431351238658605, −4.63057600504073444896097587221, −3.43948283746362730987512256110, −2.01827482783724259057924323816, 0.955094985757206389908485166138, 2.79711242316252060685475987649, 3.97332016460182174643352166363, 5.37735314378167535904351413239, 6.63567863802847576767648046217, 7.37568911914749074237357312641, 7.988294158918393362940429875132, 9.734976192664704238311977341589, 10.14850525000308014093716505442, 10.86633080798294394252442278684

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