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Properties

Label 2-380-95.49-c1-0-9
Degree $2$
Conductor $380$
Sign $-0.774 - 0.632i$
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  + (−2.10 − 1.21i)3-s + (−0.896 − 2.04i)5-s − 0.663i·7-s + (1.45 + 2.52i)9-s − 1.80·11-s + (−1.99 + 1.15i)13-s + (−0.603 + 5.40i)15-s + (3.77 + 2.18i)17-s + (−4.21 + 1.12i)19-s + (−0.806 + 1.39i)21-s + (−1.81 + 1.04i)23-s + (−3.39 + 3.67i)25-s + 0.216i·27-s + (0.974 + 1.68i)29-s − 9.52·31-s + ⋯
L(s)  = 1  + (−1.21 − 0.701i)3-s + (−0.400 − 0.916i)5-s − 0.250i·7-s + (0.485 + 0.840i)9-s − 0.545·11-s + (−0.553 + 0.319i)13-s + (−0.155 + 1.39i)15-s + (0.915 + 0.528i)17-s + (−0.966 + 0.257i)19-s + (−0.176 + 0.304i)21-s + (−0.378 + 0.218i)23-s + (−0.678 + 0.734i)25-s + 0.0416i·27-s + (0.180 + 0.313i)29-s − 1.71·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.774 - 0.632i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.774 - 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0470151 + 0.131872i\)
\(L(\frac12)\) \(\approx\) \(0.0470151 + 0.131872i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.896 + 2.04i)T \)
19 \( 1 + (4.21 - 1.12i)T \)
good3 \( 1 + (2.10 + 1.21i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + 0.663iT - 7T^{2} \)
11 \( 1 + 1.80T + 11T^{2} \)
13 \( 1 + (1.99 - 1.15i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.77 - 2.18i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.81 - 1.04i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.974 - 1.68i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.52T + 31T^{2} \)
37 \( 1 + 2.97iT - 37T^{2} \)
41 \( 1 + (0.247 - 0.428i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.81 + 3.93i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.69 + 3.28i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.99 - 1.15i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.88 + 6.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.36 + 9.28i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.96 - 2.29i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.95 - 5.12i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.86 - 2.80i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.99 - 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.20iT - 83T^{2} \)
89 \( 1 + (6.65 + 11.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.80 + 5.08i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.98274443534007831243573148631, −10.09737099572610926727205004801, −8.848757127648844401136933817675, −7.80671943571863126885995834856, −7.01952786335891260073852850453, −5.80517895089577710049221950951, −5.13698824227386351897729653588, −3.88900575550033213541481173408, −1.68342370282315517606062473142, −0.10608152359283939629935958254, 2.69994968101884903226148800544, 4.08549535062064024043736979931, 5.20084756599284850669856325259, 6.01245019565096508603976655972, 7.09338714295945418943334621143, 8.061378931870304542991207746626, 9.507340170437555619128347911458, 10.43896150726540553629854866949, 10.81501601694884199154037133466, 11.80415061911390980795224256444

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