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Properties

Label 2-380-20.7-c1-0-12
Degree $2$
Conductor $380$
Sign $-0.685 - 0.728i$
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  + (−1.34 + 0.436i)2-s + (1.71 + 1.71i)3-s + (1.61 − 1.17i)4-s + (−0.923 + 2.03i)5-s + (−3.06 − 1.56i)6-s + (0.323 − 0.323i)7-s + (−1.66 + 2.28i)8-s + 2.90i·9-s + (0.353 − 3.14i)10-s + 0.665i·11-s + (4.80 + 0.766i)12-s + (−3.14 + 3.14i)13-s + (−0.293 + 0.575i)14-s + (−5.08 + 1.91i)15-s + (1.24 − 3.80i)16-s + (1.48 + 1.48i)17-s + ⋯
L(s)  = 1  + (−0.951 + 0.308i)2-s + (0.992 + 0.992i)3-s + (0.809 − 0.586i)4-s + (−0.412 + 0.910i)5-s + (−1.25 − 0.637i)6-s + (0.122 − 0.122i)7-s + (−0.589 + 0.807i)8-s + 0.969i·9-s + (0.111 − 0.993i)10-s + 0.200i·11-s + (1.38 + 0.221i)12-s + (−0.872 + 0.872i)13-s + (−0.0784 + 0.153i)14-s + (−1.31 + 0.494i)15-s + (0.311 − 0.950i)16-s + (0.359 + 0.359i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 - 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.685 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.685 - 0.728i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.685 - 0.728i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.419450 + 0.970755i\)
\(L(\frac12)\) \(\approx\) \(0.419450 + 0.970755i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 - 0.436i)T \)
5 \( 1 + (0.923 - 2.03i)T \)
19 \( 1 + T \)
good3 \( 1 + (-1.71 - 1.71i)T + 3iT^{2} \)
7 \( 1 + (-0.323 + 0.323i)T - 7iT^{2} \)
11 \( 1 - 0.665iT - 11T^{2} \)
13 \( 1 + (3.14 - 3.14i)T - 13iT^{2} \)
17 \( 1 + (-1.48 - 1.48i)T + 17iT^{2} \)
23 \( 1 + (-4.04 - 4.04i)T + 23iT^{2} \)
29 \( 1 - 1.94iT - 29T^{2} \)
31 \( 1 + 7.94iT - 31T^{2} \)
37 \( 1 + (2.86 + 2.86i)T + 37iT^{2} \)
41 \( 1 - 0.378T + 41T^{2} \)
43 \( 1 + (-6.48 - 6.48i)T + 43iT^{2} \)
47 \( 1 + (-0.849 + 0.849i)T - 47iT^{2} \)
53 \( 1 + (7.26 - 7.26i)T - 53iT^{2} \)
59 \( 1 - 0.564T + 59T^{2} \)
61 \( 1 + 6.86T + 61T^{2} \)
67 \( 1 + (-11.3 + 11.3i)T - 67iT^{2} \)
71 \( 1 + 0.795iT - 71T^{2} \)
73 \( 1 + (-8.57 + 8.57i)T - 73iT^{2} \)
79 \( 1 - 9.50T + 79T^{2} \)
83 \( 1 + (-8.07 - 8.07i)T + 83iT^{2} \)
89 \( 1 + 16.8iT - 89T^{2} \)
97 \( 1 + (-2.53 - 2.53i)T + 97iT^{2} \)
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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

−11.20324381749063769706093947700, −10.59304747350909254642247053905, −9.557163999039794788435797427291, −9.246755530515928174998418807718, −7.952595180408876689185082622565, −7.38968636705000367612314756962, −6.24379978914943153682916610368, −4.64781939414803707194353668549, −3.39886130279132125013491497582, −2.27035476406557126893935542623, 0.866483556488401313823870552673, 2.27801289344887496554151290649, 3.36799986051654236970471056752, 5.12862015164826496854544280901, 6.80134384378372818682049745070, 7.60492499255018696798793857599, 8.342818423283712563781259970319, 8.846525944279659233068559548799, 9.871867409020361767326949203514, 10.97795519794811891693098871903

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