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Properties

Label 2-380-95.94-c0-0-1
Degree $2$
Conductor $380$
Sign $0.5 + 0.866i$
Analytic cond. $0.189644$
Root an. cond. $0.435482$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)5-s − 1.73i·7-s − 9-s + 11-s + 1.73i·17-s + 19-s + (−0.499 + 0.866i)25-s + (−1.49 + 0.866i)35-s − 1.73i·43-s + (0.5 + 0.866i)45-s + 1.73i·47-s − 1.99·49-s + (−0.5 − 0.866i)55-s + 61-s + 1.73i·63-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s − 1.73i·7-s − 9-s + 11-s + 1.73i·17-s + 19-s + (−0.499 + 0.866i)25-s + (−1.49 + 0.866i)35-s − 1.73i·43-s + (0.5 + 0.866i)45-s + 1.73i·47-s − 1.99·49-s + (−0.5 − 0.866i)55-s + 61-s + 1.73i·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.5 + 0.866i$
Analytic conductor: \(0.189644\)
Root analytic conductor: \(0.435482\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :0),\ 0.5 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7475884711\)
\(L(\frac12)\) \(\approx\) \(0.7475884711\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 - T \)
good3 \( 1 + T^{2} \)
7 \( 1 + 1.73iT - T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - 1.73iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.73iT - T^{2} \)
47 \( 1 - 1.73iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.73iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−11.39448347991777191910497595919, −10.63395831016736958726213973443, −9.564525582866044905900922091310, −8.559554467746743853534006034564, −7.79803134088040807684303816902, −6.77388089486647289399419825958, −5.57261185600090607275773745635, −4.23134329894390238626177522301, −3.60365775300727906009456452297, −1.22496783002337532100487424725, 2.51680543350416782780194495462, 3.31492373852692772330478789224, 5.04850928271146860858573829666, 5.99016608629263259011110941332, 6.94388774149781126870978416758, 8.101267339385240330948258433041, 9.043479528725945405742207084892, 9.680940375097192819510039946259, 11.21607423167003868914747075071, 11.73940428663520134005828723224

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