↑

Properties

Label 2-378-189.38-c1-0-15
Degree $2$
Conductor $378$
Sign $-0.272 + 0.962i$
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  + (−0.984 − 0.173i)2-s + (−0.507 + 1.65i)3-s + (0.939 + 0.342i)4-s + (−0.693 − 3.93i)5-s + (0.787 − 1.54i)6-s + (1.74 + 1.98i)7-s + (−0.866 − 0.5i)8-s + (−2.48 − 1.68i)9-s + 3.99i·10-s + (−1.98 − 0.350i)11-s + (−1.04 + 1.38i)12-s + (−2.86 − 3.41i)13-s + (−1.37 − 2.25i)14-s + (6.86 + 0.848i)15-s + (0.766 + 0.642i)16-s − 3.82·17-s + ⋯
L(s)  = 1  + (−0.696 − 0.122i)2-s + (−0.293 + 0.956i)3-s + (0.469 + 0.171i)4-s + (−0.310 − 1.75i)5-s + (0.321 − 0.629i)6-s + (0.660 + 0.750i)7-s + (−0.306 − 0.176i)8-s + (−0.828 − 0.560i)9-s + 1.26i·10-s + (−0.598 − 0.105i)11-s + (−0.301 + 0.399i)12-s + (−0.793 − 0.945i)13-s + (−0.368 − 0.603i)14-s + (1.77 + 0.219i)15-s + (0.191 + 0.160i)16-s − 0.928·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.272 + 0.962i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.272 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.308567 - 0.408272i\)
\(L(\frac12)\) \(\approx\) \(0.308567 - 0.408272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.984 + 0.173i)T \)
3 \( 1 + (0.507 - 1.65i)T \)
7 \( 1 + (-1.74 - 1.98i)T \)
good5 \( 1 + (0.693 + 3.93i)T + (-4.69 + 1.71i)T^{2} \)
11 \( 1 + (1.98 + 0.350i)T + (10.3 + 3.76i)T^{2} \)
13 \( 1 + (2.86 + 3.41i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + 3.82T + 17T^{2} \)
19 \( 1 + 1.11iT - 19T^{2} \)
23 \( 1 + (0.693 + 0.825i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (-5.94 + 7.08i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (-3.57 + 9.81i)T + (-23.7 - 19.9i)T^{2} \)
37 \( 1 + (-1.80 + 3.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.0624 - 0.0523i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (9.44 - 3.43i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (2.71 - 0.986i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-3.05 - 1.76i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.18 - 3.51i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (2.76 + 7.59i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.688 - 3.90i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (2.43 - 1.40i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-11.4 + 6.63i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.34 - 13.3i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-1.67 - 1.40i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + 0.880T + 89T^{2} \)
97 \( 1 + (-2.88 - 7.92i)T + (-74.3 + 62.3i)T^{2} \)
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

−11.13096785383666323468706109444, −9.934686529072843656918107189101, −9.316927679174145063015773881549, −8.312439216522560670173926098611, −8.031552262042096690258792183366, −5.98624424427361301488556404040, −5.04680134034528482882577079476, −4.38189765578063597936351716107, −2.49804320836589648881427727104, −0.42426382965529342129897707929, 1.87910573477038098803737307765, 3.05269995327675709343151387298, 4.89230843035313067332096461646, 6.62596592478186981421378761129, 6.87153736825105224262356810858, 7.67114147288174766641966710446, 8.562519528824924290154042974189, 10.19885852164221885169553559605, 10.67495420592121810846435406297, 11.47072022841617387022824609237

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