Properties

Label 2-380-19.17-c1-0-3
Degree $2$
Conductor $380$
Sign $0.759 - 0.650i$
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  + (1.21 + 1.01i)3-s + (0.939 − 0.342i)5-s + (1.77 + 3.08i)7-s + (−0.0870 − 0.493i)9-s + (1.71 − 2.97i)11-s + (−2.46 + 2.06i)13-s + (1.48 + 0.540i)15-s + (−0.843 + 4.78i)17-s + (4.05 − 1.59i)19-s + (−0.976 + 5.54i)21-s + (−6.41 − 2.33i)23-s + (0.766 − 0.642i)25-s + (2.76 − 4.79i)27-s + (0.331 + 1.88i)29-s + (1.52 + 2.63i)31-s + ⋯
L(s)  = 1  + (0.699 + 0.586i)3-s + (0.420 − 0.152i)5-s + (0.672 + 1.16i)7-s + (−0.0290 − 0.164i)9-s + (0.518 − 0.897i)11-s + (−0.682 + 0.572i)13-s + (0.383 + 0.139i)15-s + (−0.204 + 1.16i)17-s + (0.930 − 0.366i)19-s + (−0.213 + 1.20i)21-s + (−1.33 − 0.486i)23-s + (0.153 − 0.128i)25-s + (0.532 − 0.922i)27-s + (0.0616 + 0.349i)29-s + (0.273 + 0.473i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.759 - 0.650i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.759 - 0.650i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76134 + 0.651387i\)
\(L(\frac12)\) \(\approx\) \(1.76134 + 0.651387i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (-4.05 + 1.59i)T \)
good3 \( 1 + (-1.21 - 1.01i)T + (0.520 + 2.95i)T^{2} \)
7 \( 1 + (-1.77 - 3.08i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.71 + 2.97i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.46 - 2.06i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.843 - 4.78i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (6.41 + 2.33i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.331 - 1.88i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-1.52 - 2.63i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.81T + 37T^{2} \)
41 \( 1 + (6.22 + 5.22i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (10.6 - 3.86i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.899 + 5.10i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-6.97 - 2.53i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-1.53 + 8.68i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (7.19 + 2.61i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.15 - 6.56i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-7.66 + 2.78i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (5.87 + 4.92i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (10.4 + 8.74i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (8.21 + 14.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.74 + 2.30i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-0.167 + 0.952i)T + (-91.1 - 33.1i)T^{2} \)
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

−11.68064884597058630951362519660, −10.33999471651715493771641612201, −9.468806187975458949683879998116, −8.715647666370520470686488589685, −8.202167511067192696355429726959, −6.54352762515872116520058797278, −5.61875635400539761965438239493, −4.46457884469724119631168814185, −3.22081611429433729669044246485, −1.94588925754256785870490024158, 1.48090172195164574813247083794, 2.71816841293060478225656665569, 4.23893830937645674033581841970, 5.31158098965918205546728797477, 6.87922671368898318916258077211, 7.54538494085673932481821262443, 8.160022074247270906337086471046, 9.649256988410074656133502166500, 10.07534719456480150537053979261, 11.33319504216620989533058928927

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