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Properties

Label 2-380-76.31-c1-0-14
Degree $2$
Conductor $380$
Sign $-0.259 - 0.965i$
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  + (1.05 + 0.945i)2-s + (0.256 + 0.445i)3-s + (0.211 + 1.98i)4-s + (0.5 + 0.866i)5-s + (−0.150 + 0.711i)6-s + 1.09i·7-s + (−1.65 + 2.29i)8-s + (1.36 − 2.36i)9-s + (−0.293 + 1.38i)10-s + 3.04i·11-s + (−0.830 + 0.605i)12-s + (−1.26 − 0.732i)13-s + (−1.03 + 1.14i)14-s + (−0.256 + 0.445i)15-s + (−3.91 + 0.839i)16-s + (−2.28 − 3.95i)17-s + ⋯
L(s)  = 1  + (0.743 + 0.668i)2-s + (0.148 + 0.256i)3-s + (0.105 + 0.994i)4-s + (0.223 + 0.387i)5-s + (−0.0615 + 0.290i)6-s + 0.412i·7-s + (−0.586 + 0.809i)8-s + (0.455 − 0.789i)9-s + (−0.0927 + 0.437i)10-s + 0.916i·11-s + (−0.239 + 0.174i)12-s + (−0.351 − 0.203i)13-s + (−0.275 + 0.306i)14-s + (−0.0663 + 0.114i)15-s + (−0.977 + 0.209i)16-s + (−0.554 − 0.960i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.259 - 0.965i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.259 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29108 + 1.68321i\)
\(L(\frac12)\) \(\approx\) \(1.29108 + 1.68321i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.05 - 0.945i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-4.20 - 1.15i)T \)
good3 \( 1 + (-0.256 - 0.445i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 1.09iT - 7T^{2} \)
11 \( 1 - 3.04iT - 11T^{2} \)
13 \( 1 + (1.26 + 0.732i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.28 + 3.95i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.879 - 0.507i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.38 - 2.53i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.37T + 31T^{2} \)
37 \( 1 + 8.84iT - 37T^{2} \)
41 \( 1 + (-0.478 + 0.276i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.50 + 4.33i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.0276 - 0.0159i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.91 - 1.10i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.46 + 4.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.84 + 3.20i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.70 - 6.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.153 - 0.266i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.89 + 5.01i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.197 - 0.341i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.8iT - 83T^{2} \)
89 \( 1 + (-12.0 - 6.95i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.00683 + 0.00394i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.00499049627443654985654787688, −10.78200312079132174514636551741, −9.543671228772707818738621263451, −8.971387587908370348469909780862, −7.44523511879661546174459424050, −6.99418980537402096311076893635, −5.78015594540940524744696853754, −4.81365775917069826048135555339, −3.67382727256979205074555290911, −2.47715669918802642112580395403, 1.28913647978364731496064856659, 2.67190879003600863965990663999, 4.05544240490593623475725310394, 5.02342277310335158570519513795, 6.07385274489344226419294592087, 7.18632736943438966254045266917, 8.379328643943098416220860741690, 9.461308180924666322248693178476, 10.43171939869296979287781162836, 11.09032256793759301096974981663

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