↑

Properties

Label 2-378-63.4-c1-0-4
Degree $2$
Conductor $378$
Sign $0.384 + 0.923i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s − 1.58·5-s + (−2.64 − 0.0963i)7-s + 0.999·8-s + (0.794 − 1.37i)10-s + 1.58·11-s + (2.40 − 4.16i)13-s + (1.40 − 2.24i)14-s + (−0.5 + 0.866i)16-s + (2.69 − 4.67i)17-s + (−3.54 − 6.14i)19-s + (0.794 + 1.37i)20-s + (−0.794 + 1.37i)22-s − 0.300·23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s − 0.710·5-s + (−0.999 − 0.0364i)7-s + 0.353·8-s + (0.251 − 0.434i)10-s + 0.478·11-s + (0.667 − 1.15i)13-s + (0.375 − 0.599i)14-s + (−0.125 + 0.216i)16-s + (0.654 − 1.13i)17-s + (−0.814 − 1.41i)19-s + (0.177 + 0.307i)20-s + (−0.169 + 0.293i)22-s − 0.0626·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.384 + 0.923i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.384 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.519194 - 0.346064i\)
\(L(\frac12)\) \(\approx\) \(0.519194 - 0.346064i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (2.64 + 0.0963i)T \)
good5 \( 1 + 1.58T + 5T^{2} \)
11 \( 1 - 1.58T + 11T^{2} \)
13 \( 1 + (-2.40 + 4.16i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.69 + 4.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.54 + 6.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.300T + 23T^{2} \)
29 \( 1 + (4.13 + 7.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.35 - 2.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.93 - 5.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.833 + 1.44i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.33 + 2.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.44 - 4.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.23 - 5.60i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.23 + 3.87i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.02 - 8.70i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + (-8.02 + 13.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.19 - 7.26i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.18 + 2.04i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.60 + 2.78i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.712 - 1.23i)T + (-48.5 + 84.0i)T^{2} \)
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

−11.14733736667289451866259235613, −10.08938640986445483370144474859, −9.297056691059508565107549911203, −8.338357570662908724774989279574, −7.42667138031756938272507302555, −6.55167411460536584485989717777, −5.55940084735348107065927736712, −4.19174952634560981867959741388, −2.99197892063502348043752662188, −0.47917294819184509109412475082, 1.72543276160858166446234379667, 3.60050624706311344299279063644, 3.99317401508695538031094859079, 5.89381929066827786350808604127, 6.84660254686393453745340411892, 8.040453910574620024323675229422, 8.847915687339621451727062839948, 9.769661439535378098621529712381, 10.61904579019137042977631485427, 11.54317444974217256483310657108

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