↑

Properties

Label 2-378-63.32-c2-0-12
Degree $2$
Conductor $378$
Sign $0.672 + 0.740i$
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  + (1.22 + 0.707i)2-s + (0.999 + 1.73i)4-s − 2.72i·5-s + (−4.37 − 5.46i)7-s + 2.82i·8-s + (1.92 − 3.34i)10-s − 2.91i·11-s + (10.4 − 18.1i)13-s + (−1.49 − 9.78i)14-s + (−2.00 + 3.46i)16-s + (24.7 + 14.2i)17-s + (−17.7 − 30.7i)19-s + (4.72 − 2.72i)20-s + (2.05 − 3.56i)22-s − 15.7i·23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s − 0.545i·5-s + (−0.625 − 0.780i)7-s + 0.353i·8-s + (0.192 − 0.334i)10-s − 0.264i·11-s + (0.805 − 1.39i)13-s + (−0.106 − 0.698i)14-s + (−0.125 + 0.216i)16-s + (1.45 + 0.841i)17-s + (−0.933 − 1.61i)19-s + (0.236 − 0.136i)20-s + (0.0935 − 0.162i)22-s − 0.683i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.672 + 0.740i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ 0.672 + 0.740i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.96352 - 0.868848i\)
\(L(\frac12)\) \(\approx\) \(1.96352 - 0.868848i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 \)
7 \( 1 + (4.37 + 5.46i)T \)
good5 \( 1 + 2.72iT - 25T^{2} \)
11 \( 1 + 2.91iT - 121T^{2} \)
13 \( 1 + (-10.4 + 18.1i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-24.7 - 14.2i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (17.7 + 30.7i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + 15.7iT - 529T^{2} \)
29 \( 1 + (18.1 - 10.4i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-6.23 - 10.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (5.80 + 10.0i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (26.4 + 15.2i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (12.6 + 21.9i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-73.2 - 42.3i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-15.1 - 8.77i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-37.3 + 21.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (15.1 - 26.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (43.2 + 74.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 1.24iT - 5.04e3T^{2} \)
73 \( 1 + (6.48 - 11.2i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (51.7 - 89.6i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (35.2 - 20.3i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-15.5 + 8.95i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (2.62 + 4.54i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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

−10.83574313222105950391957466359, −10.41979145725124369497966096191, −8.973911960298864095034293148601, −8.179593533011377887358159090620, −7.13487315580375178950275202150, −6.12746803929743389486234364778, −5.19925298575087546459693774644, −3.98027368228849992650288501474, −3.01485620298218246715461451535, −0.821884066170982508169982944307, 1.77483974558101890276843809938, 3.10771155281932884642197636138, 4.06470756908997421353277237157, 5.54253139498269630207432115541, 6.29351699021898562889059767757, 7.27712766525416826939744376354, 8.631484907163367830395115248892, 9.662568395369932964425290249489, 10.35174624658183730895896386613, 11.58237952603574149197288630332

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