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Properties

Label 2-380-380.379-c1-0-42
Degree $2$
Conductor $380$
Sign $-0.599 + 0.800i$
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  + (−1.27 + 0.620i)2-s − 0.701i·3-s + (1.22 − 1.57i)4-s + (−1.10 − 1.94i)5-s + (0.435 + 0.891i)6-s − 1.18·7-s + (−0.581 + 2.76i)8-s + 2.50·9-s + (2.61 + 1.78i)10-s − 1.72i·11-s + (−1.10 − 0.862i)12-s − 2.64·13-s + (1.50 − 0.736i)14-s + (−1.36 + 0.775i)15-s + (−0.980 − 3.87i)16-s − 4.62i·17-s + ⋯
L(s)  = 1  + (−0.898 + 0.439i)2-s − 0.405i·3-s + (0.614 − 0.789i)4-s + (−0.493 − 0.869i)5-s + (0.177 + 0.364i)6-s − 0.448·7-s + (−0.205 + 0.978i)8-s + 0.835·9-s + (0.825 + 0.564i)10-s − 0.518i·11-s + (−0.319 − 0.249i)12-s − 0.733·13-s + (0.402 − 0.196i)14-s + (−0.352 + 0.200i)15-s + (−0.245 − 0.969i)16-s − 1.12i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.599 + 0.800i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.599 + 0.800i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.227929 - 0.455427i\)
\(L(\frac12)\) \(\approx\) \(0.227929 - 0.455427i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.27 - 0.620i)T \)
5 \( 1 + (1.10 + 1.94i)T \)
19 \( 1 + (2.08 - 3.82i)T \)
good3 \( 1 + 0.701iT - 3T^{2} \)
7 \( 1 + 1.18T + 7T^{2} \)
11 \( 1 + 1.72iT - 11T^{2} \)
13 \( 1 + 2.64T + 13T^{2} \)
17 \( 1 + 4.62iT - 17T^{2} \)
23 \( 1 + 6.62T + 23T^{2} \)
29 \( 1 + 7.53iT - 29T^{2} \)
31 \( 1 + 10.9T + 31T^{2} \)
37 \( 1 - 8.43T + 37T^{2} \)
41 \( 1 + 7.40iT - 41T^{2} \)
43 \( 1 - 9.29T + 43T^{2} \)
47 \( 1 + 4.10T + 47T^{2} \)
53 \( 1 + 1.98T + 53T^{2} \)
59 \( 1 + 3.13T + 59T^{2} \)
61 \( 1 - 3.75T + 61T^{2} \)
67 \( 1 + 9.50iT - 67T^{2} \)
71 \( 1 - 9.11T + 71T^{2} \)
73 \( 1 - 6.47iT - 73T^{2} \)
79 \( 1 - 6.38T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 - 14.1iT - 89T^{2} \)
97 \( 1 - 3.39T + 97T^{2} \)
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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

−10.94226625593899019456783269155, −9.728307125568895402666381397678, −9.324983878296387812149283785748, −7.971380279836266605717249921562, −7.61851846688071521591441828552, −6.44220283704872081870876139239, −5.41308634644877358813607869198, −4.06257065373566808266853548353, −2.07344453887985277462457521134, −0.43595384325984886863961389191, 2.07810618224780969977287869584, 3.43831859692021025813807260972, 4.37501406323874440324284073121, 6.33330240891490209455115001224, 7.19513735830543685824891111840, 7.905769392383222986065258681436, 9.214841677920689640592696800107, 9.935241167518100033611492773244, 10.63459325694852449099448149872, 11.32309136674384656432867748190

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