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Properties

Label 2-380-95.18-c1-0-8
Degree $2$
Conductor $380$
Sign $0.533 + 0.845i$
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  + (1.63 − 1.63i)3-s + (2.17 − 0.539i)5-s + (2.17 − 2.17i)7-s − 2.36i·9-s − 1.70·11-s + (−4.43 + 4.43i)13-s + (2.67 − 4.43i)15-s + (−1.63 + 1.63i)17-s + (−2.80 + 3.34i)19-s − 7.11i·21-s + (3.24 + 3.24i)23-s + (4.41 − 2.34i)25-s + (1.03 + 1.03i)27-s − 3.27·29-s − 7.11i·31-s + ⋯
L(s)  = 1  + (0.945 − 0.945i)3-s + (0.970 − 0.241i)5-s + (0.820 − 0.820i)7-s − 0.789i·9-s − 0.515·11-s + (−1.23 + 1.23i)13-s + (0.689 − 1.14i)15-s + (−0.395 + 0.395i)17-s + (−0.642 + 0.766i)19-s − 1.55i·21-s + (0.677 + 0.677i)23-s + (0.883 − 0.468i)25-s + (0.198 + 0.198i)27-s − 0.608·29-s − 1.27i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.533 + 0.845i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.533 + 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82950 - 1.00873i\)
\(L(\frac12)\) \(\approx\) \(1.82950 - 1.00873i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 + 0.539i)T \)
19 \( 1 + (2.80 - 3.34i)T \)
good3 \( 1 + (-1.63 + 1.63i)T - 3iT^{2} \)
7 \( 1 + (-2.17 + 2.17i)T - 7iT^{2} \)
11 \( 1 + 1.70T + 11T^{2} \)
13 \( 1 + (4.43 - 4.43i)T - 13iT^{2} \)
17 \( 1 + (1.63 - 1.63i)T - 17iT^{2} \)
23 \( 1 + (-3.24 - 3.24i)T + 23iT^{2} \)
29 \( 1 + 3.27T + 29T^{2} \)
31 \( 1 + 7.11iT - 31T^{2} \)
37 \( 1 + (-0.128 - 0.128i)T + 37iT^{2} \)
41 \( 1 + 3.83iT - 41T^{2} \)
43 \( 1 + (5.24 + 5.24i)T + 43iT^{2} \)
47 \( 1 + (-0.908 + 0.908i)T - 47iT^{2} \)
53 \( 1 + (5.47 - 5.47i)T - 53iT^{2} \)
59 \( 1 - 13.9T + 59T^{2} \)
61 \( 1 - 2.63T + 61T^{2} \)
67 \( 1 + (-7.71 - 7.71i)T + 67iT^{2} \)
71 \( 1 - 1.51iT - 71T^{2} \)
73 \( 1 + (8.70 + 8.70i)T + 73iT^{2} \)
79 \( 1 + 4.78T + 79T^{2} \)
83 \( 1 + (-6.46 - 6.46i)T + 83iT^{2} \)
89 \( 1 - 5.34T + 89T^{2} \)
97 \( 1 + (10.2 + 10.2i)T + 97iT^{2} \)
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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

−11.20138601537097900021312620240, −10.17013838276490046257850512959, −9.258673665804726829430570670941, −8.341803984598576414667270810280, −7.46937621592498169633206305615, −6.76831977246114527840980865877, −5.35575707197741706618815531375, −4.17664389692818276480228662803, −2.36676484892066764947505777786, −1.65009205616425761685337849728, 2.34470048945958086119781930426, 2.98255843005958152467255974810, 4.81864325926823400218198100577, 5.28257334145208901089879508814, 6.80061731572533177097869037137, 8.132431812101473196515583633122, 8.824667960017656533598570091848, 9.672428583222277020929885632809, 10.36042841763280697128744708232, 11.21869861965366844761449811492

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