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Properties

Label 2-380-20.3-c1-0-24
Degree $2$
Conductor $380$
Sign $-0.0898 + 0.995i$
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 − i)2-s + (−1 + i)3-s + 2i·4-s + (−2 + i)5-s + 2·6-s + (−2 − 2i)7-s + (2 − 2i)8-s + i·9-s + (3 + i)10-s + (−2 − 2i)12-s + 4i·14-s + (1 − 3i)15-s − 4·16-s + (5 − 5i)17-s + (1 − i)18-s − 19-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.577 + 0.577i)3-s + i·4-s + (−0.894 + 0.447i)5-s + 0.816·6-s + (−0.755 − 0.755i)7-s + (0.707 − 0.707i)8-s + 0.333i·9-s + (0.948 + 0.316i)10-s + (−0.577 − 0.577i)12-s + 1.06i·14-s + (0.258 − 0.774i)15-s − 16-s + (1.21 − 1.21i)17-s + (0.235 − 0.235i)18-s − 0.229·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0898 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0898 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.0898 + 0.995i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.0898 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.275250 - 0.301186i\)
\(L(\frac12)\) \(\approx\) \(0.275250 - 0.301186i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
5 \( 1 + (2 - i)T \)
19 \( 1 + T \)
good3 \( 1 + (1 - i)T - 3iT^{2} \)
7 \( 1 + (2 + 2i)T + 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (-5 + 5i)T - 17iT^{2} \)
23 \( 1 + (-4 + 4i)T - 23iT^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + (6 - 6i)T - 43iT^{2} \)
47 \( 1 + (2 + 2i)T + 47iT^{2} \)
53 \( 1 + (10 + 10i)T + 53iT^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + (-3 - 3i)T + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-5 - 5i)T + 73iT^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + (-4 + 4i)T - 83iT^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 + (10 - 10i)T - 97iT^{2} \)
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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

−11.15639221790371607001138036526, −10.11634113022597540682281333749, −9.803573576984259043241855171986, −8.340766529103677760043828393316, −7.49058586675796400849748712935, −6.64130475966195335784825733303, −4.88586476572418511361912691176, −3.85947417847140959498101092628, −2.85659164645844135043673340268, −0.42470211655815317100259624463, 1.25119970149498671791291730699, 3.49799430073330169701001365647, 5.19787086294410104352020598983, 6.01989445821401619339399391226, 6.91556295480652558461808133889, 7.81273941244522282525299847116, 8.766324518902652080421493177813, 9.494366261446820918221325504792, 10.66514855508434260980175923208, 11.63208236816003624210742197675

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