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Properties

Label 2-380-76.27-c1-0-2
Degree $2$
Conductor $380$
Sign $-0.414 + 0.910i$
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  + (−1.18 + 0.764i)2-s + (−1.38 + 2.39i)3-s + (0.831 − 1.81i)4-s + (−0.5 + 0.866i)5-s + (−0.185 − 3.90i)6-s + 2.18i·7-s + (0.401 + 2.79i)8-s + (−2.32 − 4.02i)9-s + (−0.0671 − 1.41i)10-s + 2.17i·11-s + (3.20 + 4.50i)12-s + (−4.38 + 2.53i)13-s + (−1.66 − 2.59i)14-s + (−1.38 − 2.39i)15-s + (−2.61 − 3.02i)16-s + (1.50 − 2.60i)17-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)2-s + (−0.798 + 1.38i)3-s + (0.415 − 0.909i)4-s + (−0.223 + 0.387i)5-s + (−0.0757 − 1.59i)6-s + 0.824i·7-s + (0.141 + 0.989i)8-s + (−0.774 − 1.34i)9-s + (−0.0212 − 0.446i)10-s + 0.655i·11-s + (0.925 + 1.30i)12-s + (−1.21 + 0.701i)13-s + (−0.445 − 0.694i)14-s + (−0.356 − 0.618i)15-s + (−0.654 − 0.756i)16-s + (0.364 − 0.631i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.414 + 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.414 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.414 + 0.910i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.414 + 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.177036 - 0.275127i\)
\(L(\frac12)\) \(\approx\) \(0.177036 - 0.275127i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.18 - 0.764i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (3.96 - 1.80i)T \)
good3 \( 1 + (1.38 - 2.39i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 2.18iT - 7T^{2} \)
11 \( 1 - 2.17iT - 11T^{2} \)
13 \( 1 + (4.38 - 2.53i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.50 + 2.60i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-4.62 + 2.66i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.07 + 4.08i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.63T + 31T^{2} \)
37 \( 1 + 1.65iT - 37T^{2} \)
41 \( 1 + (-3.52 - 2.03i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (10.2 + 5.94i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.310 + 0.179i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.75 - 5.63i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.15 - 3.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.18 + 7.24i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.10 - 7.11i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.24 - 5.62i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.49 - 4.31i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.74 - 6.48i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.827iT - 83T^{2} \)
89 \( 1 + (14.4 - 8.31i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.9 - 7.47i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.67860432119496923405993621583, −10.80585319105496907983877289294, −9.991968499569197357764700509073, −9.477041623344297512745639951294, −8.521980853744249119747672534498, −7.20309362076142946040557474407, −6.28043638338328536329265071389, −5.18250991351117065661204926137, −4.47572228387754241174749811717, −2.53041098490081974925148623470, 0.32201408716246005929852809761, 1.48013005974458610169702662087, 3.09989237411973777924189108101, 4.82677526646749924870298968879, 6.27978600860766102217859090265, 7.17192450099575047135457356495, 7.82651501205171595426940138404, 8.688552891358871859813310233707, 10.05781831863629887578666550138, 10.87481554016646446911108886870

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