↑

Properties

Label 2-378-63.11-c2-0-15
Degree $2$
Conductor $378$
Sign $-0.986 + 0.166i$
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 + (1.51 + 0.875i)5-s + (−1.24 − 6.88i)7-s + 2.82i·8-s + (1.23 − 2.14i)10-s + (−3.90 + 2.25i)11-s + (−4.80 − 8.32i)13-s + (−9.74 + 1.76i)14-s + 4.00·16-s + (−0.491 − 0.283i)17-s + (−10.8 − 18.7i)19-s + (−3.03 − 1.75i)20-s + (3.19 + 5.52i)22-s + (23.7 + 13.7i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.500·4-s + (0.303 + 0.175i)5-s + (−0.178 − 0.984i)7-s + 0.353i·8-s + (0.123 − 0.214i)10-s + (−0.355 + 0.205i)11-s + (−0.369 − 0.640i)13-s + (−0.695 + 0.125i)14-s + 0.250·16-s + (−0.0288 − 0.0166i)17-s + (−0.571 − 0.989i)19-s + (−0.151 − 0.0875i)20-s + (0.145 + 0.251i)22-s + (1.03 + 0.596i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.166i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.986 + 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.986 + 0.166i$
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: \((2,\ 378,\ (\ :1),\ -0.986 + 0.166i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0785250 - 0.935951i\)
\(L(\frac12)\) \(\approx\) \(0.0785250 - 0.935951i\)
\(L(2)\) 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 \( 1 + (1.24 + 6.88i)T \)
good5 \( 1 + (-1.51 - 0.875i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (3.90 - 2.25i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (4.80 + 8.32i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (0.491 + 0.283i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (10.8 + 18.7i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-23.7 - 13.7i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (48.9 + 28.2i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 39.9T + 961T^{2} \)
37 \( 1 + (7.44 + 12.8i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-28.0 + 16.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (1.47 - 2.55i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 - 55.1iT - 2.20e3T^{2} \)
53 \( 1 + (48.3 + 27.9i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + 12.4iT - 3.48e3T^{2} \)
61 \( 1 - 79.5T + 3.72e3T^{2} \)
67 \( 1 - 0.284T + 4.48e3T^{2} \)
71 \( 1 - 8.92iT - 5.04e3T^{2} \)
73 \( 1 + (-4.02 + 6.97i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 - 97.6T + 6.24e3T^{2} \)
83 \( 1 + (-19.5 - 11.3i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (47.1 - 27.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-38.0 + 65.8i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−10.83712666511435598201853698815, −9.888332903267634796076157428561, −9.210347161431726328838341732458, −7.85825693455815998371338656854, −7.06439505252872841987198688705, −5.70500948727693702979109294871, −4.55950724179440718568788163083, −3.43011480583168052526889297720, −2.14474175263741825671589005411, −0.39964825168287827630609192935, 1.97212194135216466522919231906, 3.57756974093021858833925796582, 5.06601843517448143322284204808, 5.75365730522564561778360277724, 6.79485191303553691647548744041, 7.82969509996113914701094877866, 8.940632016392456900527317615432, 9.359337767793743058373608315917, 10.58957019801351260050070530085, 11.63496926292545476032520613279

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