↑

Properties

Label 2-378-63.11-c2-0-5
Degree $2$
Conductor $378$
Sign $0.997 - 0.0741i$
Analytic cond. $10.2997$
Root an. cond. $3.20932$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 2.00·4-s + (7.41 + 4.28i)5-s + (−6.97 − 0.609i)7-s + 2.82i·8-s + (6.05 − 10.4i)10-s + (6.10 − 3.52i)11-s + (7.43 + 12.8i)13-s + (−0.862 + 9.86i)14-s + 4.00·16-s + (−12.4 − 7.18i)17-s + (9.66 + 16.7i)19-s + (−14.8 − 8.56i)20-s + (−4.98 − 8.63i)22-s + (34.0 + 19.6i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.500·4-s + (1.48 + 0.856i)5-s + (−0.996 − 0.0871i)7-s + 0.353i·8-s + (0.605 − 1.04i)10-s + (0.555 − 0.320i)11-s + (0.571 + 0.990i)13-s + (−0.0615 + 0.704i)14-s + 0.250·16-s + (−0.732 − 0.422i)17-s + (0.508 + 0.881i)19-s + (−0.741 − 0.428i)20-s + (−0.226 − 0.392i)22-s + (1.48 + 0.855i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0741i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.997 - 0.0741i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ 0.997 - 0.0741i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.90545 + 0.0707340i\)
\(L(\frac12)\) \(\approx\) \(1.90545 + 0.0707340i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
7 \( 1 + (6.97 + 0.609i)T \)
good5 \( 1 + (-7.41 - 4.28i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-6.10 + 3.52i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-7.43 - 12.8i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (12.4 + 7.18i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-9.66 - 16.7i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-34.0 - 19.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-11.7 - 6.80i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 24.1T + 961T^{2} \)
37 \( 1 + (-17.6 - 30.5i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (7.79 - 4.50i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-32.4 + 56.2i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + 33.3iT - 2.20e3T^{2} \)
53 \( 1 + (-52.4 - 30.2i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + 72.3iT - 3.48e3T^{2} \)
61 \( 1 + 7.98T + 3.72e3T^{2} \)
67 \( 1 + 83.6T + 4.48e3T^{2} \)
71 \( 1 + 61.0iT - 5.04e3T^{2} \)
73 \( 1 + (9.83 - 17.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 10.1T + 6.24e3T^{2} \)
83 \( 1 + (15.8 + 9.15i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (40.0 - 23.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (49.1 - 85.1i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−11.01289066325952998556612032989, −10.27417331068078160592229681450, −9.356150945219111410843949689529, −9.020676141359846543034174573715, −7.06770604029780034521034611656, −6.38251595787666203999430413995, −5.42294261877402209303704454065, −3.74918361684133691279337553139, −2.77188170531969447065968143842, −1.50004913401583900403375743949, 0.961947392099872789207958555622, 2.75861982357863089566879542527, 4.44246599648409528397594540029, 5.55404724850105511994610981999, 6.22089980587893854903078785069, 7.12127036566148182649746712905, 8.694733499359514702327620644389, 9.135002084767554977921469123004, 9.915418166860604916640867150667, 10.89866271453215300724415614496

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