Properties

Label 2-380-380.59-c1-0-48
Degree $2$
Conductor $380$
Sign $-0.907 + 0.420i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.997 + 1.00i)2-s + (0.300 − 0.357i)3-s + (−0.0105 − 1.99i)4-s + (−1.26 − 1.84i)5-s + (0.0593 + 0.658i)6-s + (−1.68 − 2.91i)7-s + (2.01 + 1.98i)8-s + (0.483 + 2.73i)9-s + (3.10 + 0.573i)10-s + (−0.107 − 0.0619i)11-s + (−0.719 − 0.597i)12-s + (−4.63 + 3.89i)13-s + (4.59 + 1.21i)14-s + (−1.03 − 0.101i)15-s + (−3.99 + 0.0421i)16-s + (−5.27 − 0.929i)17-s + ⋯
L(s)  = 1  + (−0.705 + 0.708i)2-s + (0.173 − 0.206i)3-s + (−0.00527 − 0.999i)4-s + (−0.564 − 0.825i)5-s + (0.0242 + 0.268i)6-s + (−0.635 − 1.10i)7-s + (0.712 + 0.701i)8-s + (0.161 + 0.913i)9-s + (0.983 + 0.181i)10-s + (−0.0323 − 0.0186i)11-s + (−0.207 − 0.172i)12-s + (−1.28 + 1.07i)13-s + (1.22 + 0.325i)14-s + (−0.268 − 0.0263i)15-s + (−0.999 + 0.0105i)16-s + (−1.27 − 0.225i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.907 + 0.420i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.907 + 0.420i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0334424 - 0.151555i\)
\(L(\frac12)\) \(\approx\) \(0.0334424 - 0.151555i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.997 - 1.00i)T \)
5 \( 1 + (1.26 + 1.84i)T \)
19 \( 1 + (4.35 - 0.115i)T \)
good3 \( 1 + (-0.300 + 0.357i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (1.68 + 2.91i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.107 + 0.0619i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.63 - 3.89i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (5.27 + 0.929i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (-2.96 - 1.07i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-3.07 + 0.542i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (3.24 + 5.61i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.80T + 37T^{2} \)
41 \( 1 + (-1.73 + 2.06i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (11.3 - 4.11i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (1.96 + 11.1i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-0.110 - 0.0402i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-0.756 + 4.28i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (6.90 + 2.51i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.55 + 0.274i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (2.80 - 1.02i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-4.12 + 4.91i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (8.67 + 7.27i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-3.57 - 6.18i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.78 + 3.31i)T + (-15.4 + 87.6i)T^{2} \)
97 \( 1 + (1.90 - 10.8i)T + (-91.1 - 33.1i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.79052703501166572119062177104, −9.831005125706381977074262973351, −9.032373351632343506542790866830, −8.079983558215352911395058215246, −7.23733943845153905089119693433, −6.62966901081700144600981078187, −4.93583520485186655148573921061, −4.30014444592830575312481770294, −2.01502479217430794756273199818, −0.11911661803694137012029613998, 2.54230910870937939394110415490, 3.18620121184132973113867140774, 4.52457938000850655284648934500, 6.33586743417448813188519883712, 7.13528919133632140809844245959, 8.347353107133244640824967202343, 9.075805728603557771679529477128, 9.976344027769421213814876190860, 10.71076290651895818550318478521, 11.68054355731409507446305841121

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