↑

Properties

Label 2-378-63.59-c1-0-5
Degree $2$
Conductor $378$
Sign $0.990 + 0.137i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + 3.64·5-s + (2.62 − 0.310i)7-s − 0.999i·8-s + (−3.15 − 1.82i)10-s + 5.06i·11-s + (−2.94 − 1.69i)13-s + (−2.43 − 1.04i)14-s + (−0.5 + 0.866i)16-s + (−0.774 + 1.34i)17-s + (−0.707 + 0.408i)19-s + (1.82 + 3.15i)20-s + (2.53 − 4.38i)22-s − 1.70i·23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + 1.62·5-s + (0.993 − 0.117i)7-s − 0.353i·8-s + (−0.997 − 0.576i)10-s + 1.52i·11-s + (−0.816 − 0.471i)13-s + (−0.649 − 0.279i)14-s + (−0.125 + 0.216i)16-s + (−0.187 + 0.325i)17-s + (−0.162 + 0.0936i)19-s + (0.407 + 0.705i)20-s + (0.540 − 0.935i)22-s − 0.354i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.137i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.990 + 0.137i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.990 + 0.137i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39938 - 0.0964582i\)
\(L(\frac12)\) \(\approx\) \(1.39938 - 0.0964582i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-2.62 + 0.310i)T \)
good5 \( 1 - 3.64T + 5T^{2} \)
11 \( 1 - 5.06iT - 11T^{2} \)
13 \( 1 + (2.94 + 1.69i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.774 - 1.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.707 - 0.408i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.70iT - 23T^{2} \)
29 \( 1 + (-3.60 + 2.08i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.87 - 1.08i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.39 + 5.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.01 + 1.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.06 - 5.30i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.37 + 5.83i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (11.4 + 6.63i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.08 - 1.88i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.28 + 3.62i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.22 + 2.12i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.74iT - 71T^{2} \)
73 \( 1 + (-3.76 - 2.17i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.37 - 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.768 - 1.33i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.01 + 10.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.59 + 3.23i)T + (48.5 - 84.0i)T^{2} \)
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

−11.06739909867806281982226708200, −10.19975258573574510385551945051, −9.748253740390776554026488166131, −8.777631026009858317820495678307, −7.66844091691654834759663124326, −6.73819805515516008718272485356, −5.46409038400644798399510115201, −4.50579722942426468879666940175, −2.43897566056621589157003496230, −1.68983962785756384722911910482, 1.45091043132027798179302620105, 2.68728101116147854159489052048, 4.85732533348548939135982731555, 5.71477543910783753298483223893, 6.52934491815294142819317129273, 7.74863894741070475636524676793, 8.820063645457606983778567312209, 9.351132554608751065540201665511, 10.42710807246248264172119051313, 11.08059736044971661990949869520

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