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Properties

Label 2-380-20.7-c1-0-50
Degree $2$
Conductor $380$
Sign $-0.886 - 0.462i$
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  + (−0.287 − 1.38i)2-s + (−1.15 − 1.15i)3-s + (−1.83 + 0.796i)4-s + (−0.283 − 2.21i)5-s + (−1.26 + 1.93i)6-s + (3.26 − 3.26i)7-s + (1.63 + 2.31i)8-s − 0.330i·9-s + (−2.98 + 1.03i)10-s − 4.56i·11-s + (3.04 + 1.19i)12-s + (−3.90 + 3.90i)13-s + (−5.45 − 3.57i)14-s + (−2.23 + 2.89i)15-s + (2.72 − 2.92i)16-s + (3.68 + 3.68i)17-s + ⋯
L(s)  = 1  + (−0.203 − 0.979i)2-s + (−0.667 − 0.667i)3-s + (−0.917 + 0.398i)4-s + (−0.126 − 0.991i)5-s + (−0.517 + 0.788i)6-s + (1.23 − 1.23i)7-s + (0.576 + 0.816i)8-s − 0.110i·9-s + (−0.945 + 0.325i)10-s − 1.37i·11-s + (0.877 + 0.346i)12-s + (−1.08 + 1.08i)13-s + (−1.45 − 0.956i)14-s + (−0.577 + 0.746i)15-s + (0.682 − 0.730i)16-s + (0.894 + 0.894i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.886 - 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.886 - 0.462i$
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: \((2,\ 380,\ (\ :1/2),\ -0.886 - 0.462i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.200648 + 0.819261i\)
\(L(\frac12)\) \(\approx\) \(0.200648 + 0.819261i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.287 + 1.38i)T \)
5 \( 1 + (0.283 + 2.21i)T \)
19 \( 1 + T \)
good3 \( 1 + (1.15 + 1.15i)T + 3iT^{2} \)
7 \( 1 + (-3.26 + 3.26i)T - 7iT^{2} \)
11 \( 1 + 4.56iT - 11T^{2} \)
13 \( 1 + (3.90 - 3.90i)T - 13iT^{2} \)
17 \( 1 + (-3.68 - 3.68i)T + 17iT^{2} \)
23 \( 1 + (-0.671 - 0.671i)T + 23iT^{2} \)
29 \( 1 - 1.43iT - 29T^{2} \)
31 \( 1 + 2.10iT - 31T^{2} \)
37 \( 1 + (-3.01 - 3.01i)T + 37iT^{2} \)
41 \( 1 - 3.51T + 41T^{2} \)
43 \( 1 + (-1.49 - 1.49i)T + 43iT^{2} \)
47 \( 1 + (-5.49 + 5.49i)T - 47iT^{2} \)
53 \( 1 + (-5.92 + 5.92i)T - 53iT^{2} \)
59 \( 1 + 1.07T + 59T^{2} \)
61 \( 1 + 11.0T + 61T^{2} \)
67 \( 1 + (-1.29 + 1.29i)T - 67iT^{2} \)
71 \( 1 - 1.88iT - 71T^{2} \)
73 \( 1 + (3.20 - 3.20i)T - 73iT^{2} \)
79 \( 1 + 2.07T + 79T^{2} \)
83 \( 1 + (-2.08 - 2.08i)T + 83iT^{2} \)
89 \( 1 + 10.9iT - 89T^{2} \)
97 \( 1 + (-1.72 - 1.72i)T + 97iT^{2} \)
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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

−11.14618539700135717634680544610, −10.12841435945178675544422640733, −8.993627922298229638956334074596, −8.112002036682154838223474959023, −7.33250784706205486783811253569, −5.75254728441191619658555267383, −4.68329498909732214871843530993, −3.79441561761006757906969427767, −1.63339589804080822866255246920, −0.71140880502553054854179849315, 2.45998860611492345578716675785, 4.51616514106890876252696346632, 5.18364940001561941800445689409, 5.93070246196881065565593056320, 7.49928847038278072833069843753, 7.72675278236367335972120779208, 9.204517467905345961841883325330, 10.06938486666718783478977028923, 10.71819257034397704405447844550, 11.83089053273628953437554948353

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