Properties

Label 2-380-20.19-c2-0-17
Degree $2$
Conductor $380$
Sign $-0.637 - 0.770i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.86 − 0.734i)2-s + 2.97·3-s + (2.92 + 2.73i)4-s + (−0.632 + 4.95i)5-s + (−5.52 − 2.18i)6-s − 5.57·7-s + (−3.42 − 7.23i)8-s − 0.174·9-s + (4.82 − 8.76i)10-s − 6.76i·11-s + (8.67 + 8.12i)12-s + 13.4i·13-s + (10.3 + 4.10i)14-s + (−1.87 + 14.7i)15-s + (1.05 + 15.9i)16-s + 15.9i·17-s + ⋯
L(s)  = 1  + (−0.930 − 0.367i)2-s + 0.990·3-s + (0.730 + 0.683i)4-s + (−0.126 + 0.991i)5-s + (−0.920 − 0.363i)6-s − 0.797·7-s + (−0.427 − 0.903i)8-s − 0.0194·9-s + (0.482 − 0.876i)10-s − 0.614i·11-s + (0.722 + 0.676i)12-s + 1.03i·13-s + (0.741 + 0.292i)14-s + (−0.125 + 0.982i)15-s + (0.0658 + 0.997i)16-s + 0.935i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.637 - 0.770i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ -0.637 - 0.770i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.286491 + 0.609140i\)
\(L(\frac12)\) \(\approx\) \(0.286491 + 0.609140i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.86 + 0.734i)T \)
5 \( 1 + (0.632 - 4.95i)T \)
19 \( 1 + 4.35iT \)
good3 \( 1 - 2.97T + 9T^{2} \)
7 \( 1 + 5.57T + 49T^{2} \)
11 \( 1 + 6.76iT - 121T^{2} \)
13 \( 1 - 13.4iT - 169T^{2} \)
17 \( 1 - 15.9iT - 289T^{2} \)
23 \( 1 + 37.1T + 529T^{2} \)
29 \( 1 - 45.3T + 841T^{2} \)
31 \( 1 + 10.3iT - 961T^{2} \)
37 \( 1 - 73.5iT - 1.36e3T^{2} \)
41 \( 1 + 73.2T + 1.68e3T^{2} \)
43 \( 1 + 69.3T + 1.84e3T^{2} \)
47 \( 1 - 41.1T + 2.20e3T^{2} \)
53 \( 1 + 75.3iT - 2.80e3T^{2} \)
59 \( 1 - 64.6iT - 3.48e3T^{2} \)
61 \( 1 - 52.6T + 3.72e3T^{2} \)
67 \( 1 - 40.0T + 4.48e3T^{2} \)
71 \( 1 - 2.17iT - 5.04e3T^{2} \)
73 \( 1 - 81.2iT - 5.32e3T^{2} \)
79 \( 1 - 89.5iT - 6.24e3T^{2} \)
83 \( 1 - 105.T + 6.88e3T^{2} \)
89 \( 1 + 9.02T + 7.92e3T^{2} \)
97 \( 1 - 36.2iT - 9.40e3T^{2} \)
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

−11.43460289159157212625135768744, −10.14876152471058220596421102696, −9.848338624423415330475190921379, −8.502642899884314819104972439103, −8.211688985828914382859106266858, −6.81169470101337904123148621311, −6.29409449149553014929458450426, −3.80934206975712485861439040629, −3.09504847656516122510490673651, −2.03761722238322875068920190512, 0.32792560360792953200150814822, 2.09875325300655662135565410303, 3.38049739166196840436140089018, 5.04644726316368459763036048167, 6.15433615463813318354787132207, 7.43523844816871217157787818135, 8.182639415692315012596686125930, 8.868595043969162718788792936641, 9.708848069892177125357670136919, 10.28154982268897495693024478329

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