↑

Properties

Label 2-378-189.101-c1-0-20
Degree $2$
Conductor $378$
Sign $0.234 + 0.972i$
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  + (−0.642 + 0.766i)2-s + (0.0131 − 1.73i)3-s + (−0.173 − 0.984i)4-s + (3.23 − 2.71i)5-s + (1.31 + 1.12i)6-s + (1.66 − 2.05i)7-s + (0.866 + 0.500i)8-s + (−2.99 − 0.0454i)9-s + 4.22i·10-s + (−0.952 + 1.13i)11-s + (−1.70 + 0.287i)12-s + (−0.0656 + 0.180i)13-s + (0.502 + 2.59i)14-s + (−4.66 − 5.64i)15-s + (−0.939 + 0.342i)16-s − 2.02·17-s + ⋯
L(s)  = 1  + (−0.454 + 0.541i)2-s + (0.00757 − 0.999i)3-s + (−0.0868 − 0.492i)4-s + (1.44 − 1.21i)5-s + (0.538 + 0.458i)6-s + (0.629 − 0.776i)7-s + (0.306 + 0.176i)8-s + (−0.999 − 0.0151i)9-s + 1.33i·10-s + (−0.287 + 0.342i)11-s + (−0.493 + 0.0830i)12-s + (−0.0182 + 0.0500i)13-s + (0.134 + 0.694i)14-s + (−1.20 − 1.45i)15-s + (−0.234 + 0.0855i)16-s − 0.489·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.234 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.234 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.234 + 0.972i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.234 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04787 - 0.824860i\)
\(L(\frac12)\) \(\approx\) \(1.04787 - 0.824860i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.642 - 0.766i)T \)
3 \( 1 + (-0.0131 + 1.73i)T \)
7 \( 1 + (-1.66 + 2.05i)T \)
good5 \( 1 + (-3.23 + 2.71i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (0.952 - 1.13i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.0656 - 0.180i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + 2.02T + 17T^{2} \)
19 \( 1 - 2.77iT - 19T^{2} \)
23 \( 1 + (3.18 - 8.73i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.08 + 2.97i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-8.59 + 1.51i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-0.880 + 1.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-8.23 - 2.99i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.731 + 4.15i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (1.78 - 10.1i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-0.697 - 0.402i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (12.1 + 4.43i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.731 - 0.128i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-2.31 + 1.94i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.21 - 0.703i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.52 + 2.61i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.61 - 4.71i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-3.70 + 1.34i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + (11.9 + 2.10i)T + (91.1 + 33.1i)T^{2} \)
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

−11.12572178534615114259972445323, −9.930450393340362602679031368945, −9.290718043117841596984111948050, −8.182875264142237383375038974339, −7.62106545196625572069552241017, −6.31609258620665450829450270609, −5.62711577978484450604769673679, −4.57500870911669546142840595792, −2.06246989421179891254071363337, −1.13638733855306502354979981823, 2.30589394308591105590384997829, 2.91387462176812737255612213344, 4.61985009670757196581204836783, 5.75037775653281572894942021831, 6.63434763201680805281996764036, 8.256372806547474316813431882513, 9.065876130521485700173905612582, 9.872439649919016612982358000723, 10.67213242377808284012134175852, 11.04231783299677013683612512013

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