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Properties

Label 2-380-95.64-c1-0-8
Degree $2$
Conductor $380$
Sign $0.833 + 0.551i$
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 + (2.22 + 0.248i)5-s − 0.663i·7-s + (1.45 − 2.52i)9-s − 1.80·11-s + (1.99 + 1.15i)13-s + (4.98 − 2.17i)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 + (4.87 + 1.10i)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 + (0.993 + 0.111i)5-s − 0.250i·7-s + (0.485 − 0.840i)9-s − 0.545·11-s + (0.553 + 0.319i)13-s + (1.28 − 0.562i)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 + (0.975 + 0.220i)25-s + 0.0416i·27-s + (0.180 − 0.313i)29-s − 1.71·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.833 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.833 + 0.551i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.833 + 0.551i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08808 - 0.628495i\)
\(L(\frac12)\) \(\approx\) \(2.08808 - 0.628495i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 - 0.248i)T \)
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

−11.05253470891881854782435189465, −10.36058118846403999286749943950, −9.035148931466335181556993848275, −8.744696531950349262089813472700, −7.51164420430907569169241021070, −6.70700675055409471729868417968, −5.60735839019227620504379153768, −4.03426716157377352136936476448, −2.63348939751206352004634743565, −1.76592786295930629640494553001, 2.10159418239739150371534021794, 3.07998716683768140261406548209, 4.39239642933472719546595964786, 5.51709094044718382638520913769, 6.67530199676204948986279764549, 8.063801680810099890073397383119, 8.881630220323243713367486687588, 9.406459411310919909309359048075, 10.38036763875119817925337661890, 11.08669421224391066431967933307

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