↑

Properties

Label 2-380-76.31-c1-0-18
Degree $2$
Conductor $380$
Sign $0.991 + 0.132i$
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  + (1.41 − 0.0606i)2-s + (−0.600 − 1.04i)3-s + (1.99 − 0.171i)4-s + (0.5 + 0.866i)5-s + (−0.912 − 1.43i)6-s + 4.07i·7-s + (2.80 − 0.363i)8-s + (0.777 − 1.34i)9-s + (0.759 + 1.19i)10-s − 0.526i·11-s + (−1.37 − 1.97i)12-s + (4.15 + 2.39i)13-s + (0.247 + 5.75i)14-s + (0.600 − 1.04i)15-s + (3.94 − 0.683i)16-s + (−2.96 − 5.13i)17-s + ⋯
L(s)  = 1  + (0.999 − 0.0429i)2-s + (−0.346 − 0.600i)3-s + (0.996 − 0.0857i)4-s + (0.223 + 0.387i)5-s + (−0.372 − 0.585i)6-s + 1.54i·7-s + (0.991 − 0.128i)8-s + (0.259 − 0.449i)9-s + (0.240 + 0.377i)10-s − 0.158i·11-s + (−0.397 − 0.568i)12-s + (1.15 + 0.664i)13-s + (0.0661 + 1.53i)14-s + (0.155 − 0.268i)15-s + (0.985 − 0.170i)16-s + (−0.719 − 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.991 + 0.132i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.991 + 0.132i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.38819 - 0.159167i\)
\(L(\frac12)\) \(\approx\) \(2.38819 - 0.159167i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0606i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (3.48 + 2.61i)T \)
good3 \( 1 + (0.600 + 1.04i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 4.07iT - 7T^{2} \)
11 \( 1 + 0.526iT - 11T^{2} \)
13 \( 1 + (-4.15 - 2.39i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.96 + 5.13i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.08 + 0.628i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.69 + 1.55i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.0294T + 31T^{2} \)
37 \( 1 - 6.86iT - 37T^{2} \)
41 \( 1 + (6.78 - 3.91i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.46 + 2.57i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.907 - 0.523i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.74 - 1.58i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.03 + 1.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.47 - 9.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.00 + 1.74i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.94 + 5.09i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.45 + 2.51i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.81 + 11.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.72iT - 83T^{2} \)
89 \( 1 + (8.63 + 4.98i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.0 + 7.51i)T + (48.5 - 84.0i)T^{2} \)
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

−11.67741837753624730179236657509, −10.89598896650690876983215800228, −9.479086238447316804188034924133, −8.561078483473051639106317786176, −7.07981560151311759797703816102, −6.36234288366027101683439319550, −5.74169234320881343306705603540, −4.45899587462028870733039447930, −2.98925716728791642298223926071, −1.87437786544724815058558151101, 1.68473943946300779104970514302, 3.83165275983742956893478903573, 4.18789619641185200408644674152, 5.41786182757587643543838226627, 6.34611940081593463587259573318, 7.46114121232743598660629073622, 8.404064778174301139619724010626, 10.05328336437973495244653783193, 10.73299436577856646761685475471, 11.03759336408698584613165846255

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