↑

Properties

Label 2-380-380.379-c1-0-40
Degree $2$
Conductor $380$
Sign $0.284 + 0.958i$
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  + (0.203 + 1.39i)2-s − 1.12i·3-s + (−1.91 + 0.568i)4-s − 2.23·5-s + (1.57 − 0.227i)6-s + (−1.18 − 2.56i)8-s + 1.74·9-s + (−0.453 − 3.12i)10-s − 5.95i·11-s + (0.637 + 2.15i)12-s − 6.51·13-s + 2.50i·15-s + (3.35 − 2.17i)16-s + (0.353 + 2.43i)18-s − 4.35i·19-s + (4.28 − 1.27i)20-s + ⋯
L(s)  = 1  + (0.143 + 0.989i)2-s − 0.647i·3-s + (−0.958 + 0.284i)4-s − 0.999·5-s + (0.641 − 0.0930i)6-s + (−0.418 − 0.908i)8-s + 0.580·9-s + (−0.143 − 0.989i)10-s − 1.79i·11-s + (0.184 + 0.621i)12-s − 1.80·13-s + 0.647i·15-s + (0.838 − 0.544i)16-s + (0.0832 + 0.574i)18-s − 0.999i·19-s + (0.958 − 0.284i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.284 + 0.958i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.284 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.552628 - 0.412610i\)
\(L(\frac12)\) \(\approx\) \(0.552628 - 0.412610i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.203 - 1.39i)T \)
5 \( 1 + 2.23T \)
19 \( 1 + 4.35iT \)
good3 \( 1 + 1.12iT - 3T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 5.95iT - 11T^{2} \)
13 \( 1 + 6.51T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 8.01T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 5.09T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 1.11T + 61T^{2} \)
67 \( 1 - 13.7iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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

−11.41282893574545097209046607882, −10.13482621002301017860114591497, −8.934519142819062987214226891058, −8.137180847754284803300231798302, −7.29851389808293358017030903130, −6.72268727410739596999401869200, −5.38629168447042655213273300247, −4.36477416836480403431936781412, −3.07774613099976410096988920514, −0.44213399165821938233198961366, 2.01155673305768948380695139989, 3.53468823713225290428835150343, 4.54456586301138962661342586717, 4.99511566417944054670728185183, 7.06836456638501430105747007380, 7.84377124607964252524356648543, 9.206462531863377112636874697862, 9.993327651280377243372504386484, 10.41003128640736411018545430718, 11.67616155363336911051176643863

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