Properties

Label 2-380-95.37-c1-0-4
Degree $2$
Conductor $380$
Sign $0.980 + 0.194i$
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  + (1.63 + 1.52i)5-s + (−2.42 − 2.42i)7-s − 3i·9-s + 6.50·11-s + (5.57 + 5.57i)17-s − 4.35i·19-s + (−2.35 + 2.35i)23-s + (0.362 + 4.98i)25-s + (−0.278 − 7.67i)35-s + (3.07 − 3.07i)43-s + (4.56 − 4.91i)45-s + (−8.08 − 8.08i)47-s + 4.79i·49-s + (10.6 + 9.91i)55-s − 10.8·61-s + ⋯
L(s)  = 1  + (0.732 + 0.680i)5-s + (−0.917 − 0.917i)7-s i·9-s + 1.96·11-s + (1.35 + 1.35i)17-s − 0.999i·19-s + (−0.491 + 0.491i)23-s + (0.0725 + 0.997i)25-s + (−0.0470 − 1.29i)35-s + (0.469 − 0.469i)43-s + (0.680 − 0.732i)45-s + (−1.17 − 1.17i)47-s + 0.684i·49-s + (1.43 + 1.33i)55-s − 1.38·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.980 + 0.194i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.980 + 0.194i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52766 - 0.149835i\)
\(L(\frac12)\) \(\approx\) \(1.52766 - 0.149835i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 - 1.52i)T \)
19 \( 1 + 4.35iT \)
good3 \( 1 + 3iT^{2} \)
7 \( 1 + (2.42 + 2.42i)T + 7iT^{2} \)
11 \( 1 - 6.50T + 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (-5.57 - 5.57i)T + 17iT^{2} \)
23 \( 1 + (2.35 - 2.35i)T - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-3.07 + 3.07i)T - 43iT^{2} \)
47 \( 1 + (8.08 + 8.08i)T + 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (10.9 - 10.9i)T - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (12.3 - 12.3i)T - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97iT^{2} \)
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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

−11.32056294367605166400116376779, −10.15169238241224517980873794684, −9.694908724764931845826502535605, −8.823638875541426315685472295141, −7.21444383415029771927746530174, −6.54512408024478170413551641982, −5.91334719490002346061875828805, −3.93530402533065254849289795677, −3.35238091586337869211572292441, −1.32618120389125657033179923581, 1.56497233112456890170388742194, 3.03971745525660061221728959585, 4.54571402685686780720672890729, 5.72121740520123706212165583673, 6.34574603533686815107541641514, 7.71614677315693287425431679234, 8.889379576989932670086877358349, 9.491134172202822953376577007261, 10.17234561404994808400582433972, 11.68153712955073572205292252492

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