↑

Properties

Label 2-380-76.27-c1-0-11
Degree $2$
Conductor $380$
Sign $-0.983 - 0.182i$
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  + (1.25 + 0.654i)2-s + (−1.50 + 2.60i)3-s + (1.14 + 1.64i)4-s + (−0.5 + 0.866i)5-s + (−3.59 + 2.28i)6-s + 2.59i·7-s + (0.359 + 2.80i)8-s + (−3.02 − 5.24i)9-s + (−1.19 + 0.758i)10-s − 4.32i·11-s + (−5.99 + 0.510i)12-s + (5.09 − 2.94i)13-s + (−1.70 + 3.25i)14-s + (−1.50 − 2.60i)15-s + (−1.38 + 3.75i)16-s + (−0.759 + 1.31i)17-s + ⋯
L(s)  = 1  + (0.886 + 0.462i)2-s + (−0.868 + 1.50i)3-s + (0.571 + 0.820i)4-s + (−0.223 + 0.387i)5-s + (−1.46 + 0.931i)6-s + 0.982i·7-s + (0.126 + 0.991i)8-s + (−1.00 − 1.74i)9-s + (−0.377 + 0.239i)10-s − 1.30i·11-s + (−1.73 + 0.147i)12-s + (1.41 − 0.816i)13-s + (−0.454 + 0.870i)14-s + (−0.388 − 0.672i)15-s + (−0.346 + 0.938i)16-s + (−0.184 + 0.319i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.983 - 0.182i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.983 - 0.182i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.146298 + 1.59215i\)
\(L(\frac12)\) \(\approx\) \(0.146298 + 1.59215i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.25 - 0.654i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (1.56 - 4.07i)T \)
good3 \( 1 + (1.50 - 2.60i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 2.59iT - 7T^{2} \)
11 \( 1 + 4.32iT - 11T^{2} \)
13 \( 1 + (-5.09 + 2.94i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.759 - 1.31i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-4.53 + 2.61i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (8.10 - 4.67i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.63T + 31T^{2} \)
37 \( 1 - 5.86iT - 37T^{2} \)
41 \( 1 + (-0.487 - 0.281i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.31 + 2.49i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.11 - 0.643i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.04 + 1.75i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.06 + 7.04i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.65 - 2.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.662 + 1.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.864 + 1.49i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.793 + 1.37i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.810 + 1.40i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.0iT - 83T^{2} \)
89 \( 1 + (-13.2 + 7.63i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.46 - 5.46i)T + (48.5 + 84.0i)T^{2} \)
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

−11.52231609639076114041947811839, −11.06830523549627658960831246504, −10.36073768628489956265755802339, −8.838614141028751674432501820400, −8.286963191446439279068464020295, −6.40999364841063070468952580802, −5.83735712938268475780809159719, −5.13576176574132502379592439721, −3.79292354893945918120294340954, −3.17703067219480731893174363346, 0.958099570576188070801236221909, 2.06329730305382463856551095891, 4.01197200484460701007846025249, 4.99233076210538686868723782673, 6.19519659725272592312423343320, 6.97781457867423714758965147835, 7.54440238801244679782545946428, 9.166543036536149436100960055207, 10.52390190226675991994510115811, 11.40571115360066417917472735603

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