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Properties

Label 2-380-95.64-c1-0-3
Degree $2$
Conductor $380$
Sign $0.159 - 0.987i$
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  + (−0.392 + 0.226i)3-s + (1.82 + 1.29i)5-s + 2.54i·7-s + (−1.39 + 2.42i)9-s + 2.22·11-s + (−6.08 − 3.51i)13-s + (−1.00 − 0.0937i)15-s + (−2.21 + 1.27i)17-s + (2.70 + 3.41i)19-s + (−0.575 − 0.997i)21-s + (6.95 + 4.01i)23-s + (1.65 + 4.71i)25-s − 2.62i·27-s + (−0.941 + 1.63i)29-s + 5.98·31-s + ⋯
L(s)  = 1  + (−0.226 + 0.130i)3-s + (0.816 + 0.578i)5-s + 0.961i·7-s + (−0.465 + 0.806i)9-s + 0.670·11-s + (−1.68 − 0.973i)13-s + (−0.260 − 0.0242i)15-s + (−0.537 + 0.310i)17-s + (0.620 + 0.784i)19-s + (−0.125 − 0.217i)21-s + (1.44 + 0.837i)23-s + (0.331 + 0.943i)25-s − 0.505i·27-s + (−0.174 + 0.302i)29-s + 1.07·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.159 - 0.987i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.159 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.982593 + 0.836990i\)
\(L(\frac12)\) \(\approx\) \(0.982593 + 0.836990i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.82 - 1.29i)T \)
19 \( 1 + (-2.70 - 3.41i)T \)
good3 \( 1 + (0.392 - 0.226i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 - 2.54iT - 7T^{2} \)
11 \( 1 - 2.22T + 11T^{2} \)
13 \( 1 + (6.08 + 3.51i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.21 - 1.27i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-6.95 - 4.01i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.941 - 1.63i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.98T + 31T^{2} \)
37 \( 1 + 2.86iT - 37T^{2} \)
41 \( 1 + (3.67 + 6.36i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.19 - 1.84i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.09 + 2.36i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.91 - 5.14i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.73 + 6.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.17 + 7.23i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.7 - 6.17i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.13 + 7.16i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-10.9 + 6.32i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.13 + 3.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.7iT - 83T^{2} \)
89 \( 1 + (7.19 - 12.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.04 - 2.91i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.54187351084639148090587428271, −10.57582082098498497680374022217, −9.803759798868880996196585214060, −8.965345607143127758805246766790, −7.79415931088719636406211144858, −6.75246379863322549488086077668, −5.55859390762282114965125402103, −5.09803016695201678659883454612, −3.11178698159108036318104037053, −2.11694104781087431562276545453, 0.918312973430942077327864361082, 2.65820953524621787497419120309, 4.36354824383871932232561176571, 5.14796926120361601944316470942, 6.68252821161685940816825727748, 6.95282452875253952110572851151, 8.598381644915309021936406980333, 9.434001173529308391778752990255, 9.994477077875433382823088644476, 11.33613213230139323890197032049

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