↑

Properties

Label 2-378-189.88-c1-0-0
Degree $2$
Conductor $378$
Sign $-0.999 - 0.00273i$
Analytic cond. $3.01834$
Root an. cond. $1.73733$
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  + (0.766 + 0.642i)2-s + (−0.116 − 1.72i)3-s + (0.173 + 0.984i)4-s + (−3.13 + 2.62i)5-s + (1.02 − 1.39i)6-s + (−2.40 − 1.10i)7-s + (−0.500 + 0.866i)8-s + (−2.97 + 0.400i)9-s − 4.08·10-s + (−0.798 − 0.670i)11-s + (1.68 − 0.414i)12-s + (−4.41 − 1.60i)13-s + (−1.13 − 2.39i)14-s + (4.90 + 5.10i)15-s + (−0.939 + 0.342i)16-s + 2.66·17-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (−0.0669 − 0.997i)3-s + (0.0868 + 0.492i)4-s + (−1.40 + 1.17i)5-s + (0.417 − 0.570i)6-s + (−0.908 − 0.418i)7-s + (−0.176 + 0.306i)8-s + (−0.991 + 0.133i)9-s − 1.29·10-s + (−0.240 − 0.202i)11-s + (0.485 − 0.119i)12-s + (−1.22 − 0.445i)13-s + (−0.302 − 0.639i)14-s + (1.26 + 1.31i)15-s + (−0.234 + 0.0855i)16-s + 0.646·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.999 - 0.00273i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.999 - 0.00273i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000218176 + 0.159712i\)
\(L(\frac12)\) \(\approx\) \(0.000218176 + 0.159712i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 + (0.116 + 1.72i)T \)
7 \( 1 + (2.40 + 1.10i)T \)
good5 \( 1 + (3.13 - 2.62i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (0.798 + 0.670i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (4.41 + 1.60i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 - 2.66T + 17T^{2} \)
19 \( 1 + 5.03T + 19T^{2} \)
23 \( 1 + (-7.94 - 2.89i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (2.77 - 1.01i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-1.21 - 6.91i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (3.62 - 6.28i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.16 + 2.60i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.13 + 6.46i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.299 - 1.69i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (0.967 - 1.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.188 + 0.0684i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-2.24 + 12.7i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (2.43 - 2.04i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.00678 - 0.0117i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.93 + 6.82i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.16 + 5.17i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (5.05 - 1.84i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 4.94T + 89T^{2} \)
97 \( 1 + (0.589 - 3.34i)T + (-91.1 - 33.1i)T^{2} \)
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

−12.09364495583229839701295538599, −11.08237817803594627762917425127, −10.30785174831860091202712361647, −8.633733841671294026756888662321, −7.61539879376879139209801300304, −7.09355298556695246379032449348, −6.50899938553646647090937239240, −5.08526707997335978094555083945, −3.48162412738031909266542634243, −2.87417487208138696287371649070, 0.084893684670081472645723556703, 2.81828693683133829927528958236, 4.00230055336610835333702007901, 4.68299276979006425371860597856, 5.57494685808483570396500779311, 7.09418180461034536543086856716, 8.411937108232463792072109047282, 9.203310127963231766253976357700, 9.974632340399862603895631914854, 11.10662090169802029597575803321

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