Properties

Label 2-380-380.239-c2-0-10
Degree $2$
Conductor $380$
Sign $0.413 + 0.910i$
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  + (0.691 + 1.87i)2-s + (−1.97 + 3.42i)3-s + (−3.04 + 2.59i)4-s + (−3.94 + 3.07i)5-s + (−7.78 − 1.34i)6-s − 2.56·7-s + (−6.97 − 3.91i)8-s + (−3.30 − 5.73i)9-s + (−8.49 − 5.26i)10-s + 17.7i·11-s + (−2.87 − 15.5i)12-s + (4.58 − 2.64i)13-s + (−1.77 − 4.82i)14-s + (−2.73 − 19.5i)15-s + (2.52 − 15.8i)16-s + (−1.90 − 1.09i)17-s + ⋯
L(s)  = 1  + (0.345 + 0.938i)2-s + (−0.658 + 1.14i)3-s + (−0.760 + 0.649i)4-s + (−0.788 + 0.615i)5-s + (−1.29 − 0.223i)6-s − 0.367·7-s + (−0.872 − 0.489i)8-s + (−0.367 − 0.636i)9-s + (−0.849 − 0.526i)10-s + 1.61i·11-s + (−0.239 − 1.29i)12-s + (0.352 − 0.203i)13-s + (−0.126 − 0.344i)14-s + (−0.182 − 1.30i)15-s + (0.157 − 0.987i)16-s + (−0.111 − 0.0645i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 + 0.910i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.413 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.413 + 0.910i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ 0.413 + 0.910i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.540103 - 0.348010i\)
\(L(\frac12)\) \(\approx\) \(0.540103 - 0.348010i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.691 - 1.87i)T \)
5 \( 1 + (3.94 - 3.07i)T \)
19 \( 1 + (-15.3 - 11.2i)T \)
good3 \( 1 + (1.97 - 3.42i)T + (-4.5 - 7.79i)T^{2} \)
7 \( 1 + 2.56T + 49T^{2} \)
11 \( 1 - 17.7iT - 121T^{2} \)
13 \( 1 + (-4.58 + 2.64i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (1.90 + 1.09i)T + (144.5 + 250. i)T^{2} \)
23 \( 1 + (8.21 + 14.2i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-24.5 - 42.4i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + 13.0iT - 961T^{2} \)
37 \( 1 - 53.5iT - 1.36e3T^{2} \)
41 \( 1 + (-12.6 + 21.9i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-36.7 + 63.7i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-27.9 - 48.4i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (38.6 - 22.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (58.4 + 33.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (2.07 + 3.59i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (51.0 + 88.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (35.9 + 20.7i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (41.1 + 23.7i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-63.0 - 36.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 124.T + 6.88e3T^{2} \)
89 \( 1 + (-50.2 - 86.9i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-53.4 - 30.8i)T + (4.70e3 + 8.14e3i)T^{2} \)
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

−12.07234349029234478614472744506, −10.80822183920846434722804099022, −10.06986515642634301149617595741, −9.230988309420766027469266682933, −7.927061070007801675121449048977, −7.09140423730234387079447110978, −6.13877587708335694608944318711, −4.91847823358568526222342688314, −4.26856575934392727497249252649, −3.22197360462250240906424568107, 0.32306108635667575922425058733, 1.21354101854832716593924796731, 3.02344181219770380903060619548, 4.18682951292765497465268502964, 5.57233009638343404304966412940, 6.25275714754717656846425901906, 7.61953778618325318253167588455, 8.577955116506949911326436329020, 9.497826985898691346418160771984, 10.92457616698627870421679017158

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