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Properties

Label 2-380-380.379-c1-0-28
Degree $2$
Conductor $380$
Sign $0.958 + 0.284i$
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  + (−1.13 + 0.846i)2-s − 1.52i·3-s + (0.568 − 1.91i)4-s + 2.23·5-s + (1.28 + 1.72i)6-s + (0.978 + 2.65i)8-s + 0.678·9-s + (−2.53 + 1.89i)10-s + 2.92i·11-s + (−2.92 − 0.865i)12-s + 2.42·13-s − 3.40i·15-s + (−3.35 − 2.17i)16-s + (−0.768 + 0.573i)18-s − 4.35i·19-s + (1.27 − 4.28i)20-s + ⋯
L(s)  = 1  + (−0.801 + 0.598i)2-s − 0.879i·3-s + (0.284 − 0.958i)4-s + 0.999·5-s + (0.526 + 0.704i)6-s + (0.345 + 0.938i)8-s + 0.226·9-s + (−0.801 + 0.598i)10-s + 0.883i·11-s + (−0.843 − 0.249i)12-s + 0.672·13-s − 0.879i·15-s + (−0.838 − 0.544i)16-s + (−0.181 + 0.135i)18-s − 0.999i·19-s + (0.284 − 0.958i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.958 + 0.284i$
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: \((2,\ 380,\ (\ :1/2),\ 0.958 + 0.284i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15606 - 0.167700i\)
\(L(\frac12)\) \(\approx\) \(1.15606 - 0.167700i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13 - 0.846i)T \)
5 \( 1 - 2.23T \)
19 \( 1 + 4.35iT \)
good3 \( 1 + 1.52iT - 3T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 2.92iT - 11T^{2} \)
13 \( 1 - 2.42T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 0.802T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 13.2T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 15.5T + 61T^{2} \)
67 \( 1 - 3.36iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 19.4T + 97T^{2} \)
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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

−11.09258722242526506205977427825, −10.16686700321532718643960146767, −9.420672301279501264796066601662, −8.520653605076232164788645711971, −7.39225721476345790170247423329, −6.74575845752065277937193389950, −5.92290227801592347257179685857, −4.72968551055496362703691021312, −2.35078115572950743095300862436, −1.28001129111446408026241840479, 1.49868430496705304224073561224, 3.08240191714102145056890987587, 4.13920221194879398078718334147, 5.58781781575919318971763344296, 6.66638739254721923395062096416, 8.040864360747313133976746278875, 8.969409122855739648364204766068, 9.643063962755654838725080993398, 10.46610928908444327389339241965, 10.93063070770579422995695111638

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