Properties

Label 2-380-76.7-c2-0-47
Degree $2$
Conductor $380$
Sign $-0.0244 + 0.999i$
Analytic cond. $10.3542$
Root an. cond. $3.21780$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.61 − 1.17i)2-s + (−2.81 + 1.62i)3-s + (1.23 + 3.80i)4-s + (1.11 + 1.93i)5-s + (6.46 + 0.678i)6-s − 13.4i·7-s + (2.46 − 7.60i)8-s + (0.790 − 1.36i)9-s + (0.466 − 4.44i)10-s + 15.3i·11-s + (−9.67 − 8.70i)12-s + (−4.04 + 6.99i)13-s + (−15.8 + 21.8i)14-s + (−6.29 − 3.63i)15-s + (−12.9 + 9.41i)16-s + (7.58 + 13.1i)17-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.938 + 0.542i)3-s + (0.309 + 0.950i)4-s + (0.223 + 0.387i)5-s + (1.07 + 0.113i)6-s − 1.92i·7-s + (0.308 − 0.951i)8-s + (0.0877 − 0.152i)9-s + (0.0466 − 0.444i)10-s + 1.39i·11-s + (−0.805 − 0.725i)12-s + (−0.310 + 0.538i)13-s + (−1.13 + 1.55i)14-s + (−0.419 − 0.242i)15-s + (−0.808 + 0.588i)16-s + (0.446 + 0.773i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0244 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0244 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.0244 + 0.999i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ -0.0244 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.377591 - 0.386951i\)
\(L(\frac12)\) \(\approx\) \(0.377591 - 0.386951i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.61 + 1.17i)T \)
5 \( 1 + (-1.11 - 1.93i)T \)
19 \( 1 + (7.35 + 17.5i)T \)
good3 \( 1 + (2.81 - 1.62i)T + (4.5 - 7.79i)T^{2} \)
7 \( 1 + 13.4iT - 49T^{2} \)
11 \( 1 - 15.3iT - 121T^{2} \)
13 \( 1 + (4.04 - 6.99i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-7.58 - 13.1i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (21.8 + 12.5i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-21.0 + 36.4i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + 1.73iT - 961T^{2} \)
37 \( 1 - 23.8T + 1.36e3T^{2} \)
41 \( 1 + (7.36 + 12.7i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-56.3 + 32.5i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-3.67 - 2.12i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-38.6 + 66.8i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-36.3 + 20.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-8.66 + 15.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (81.2 + 46.8i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-83.5 + 48.2i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (27.9 + 48.4i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (38.8 - 22.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 82.3iT - 6.88e3T^{2} \)
89 \( 1 + (-42.9 + 74.3i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-5.58 - 9.67i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.63553275622348603118761881484, −10.20872668685151805040924372066, −9.636758765580126370930933892369, −8.030363361150302945204241957435, −7.18179248631261184765968502975, −6.41111851291334922908805618519, −4.54653946482211523908677235784, −4.02715485873850077438301133060, −2.13275015179578779650279499622, −0.41260916988608195329711792747, 1.11995976017157466601126452251, 2.71654968863913879360269382524, 5.30795239130068883245102579291, 5.75156534975692218904790370153, 6.31546614912814941890396964909, 7.75868651458007239971128604928, 8.639290119599684859150712209136, 9.264504556401819998309815472458, 10.40900167061203375559187347014, 11.53153426017134465660422588976

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