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Properties

Label 2-378-63.47-c1-0-2
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

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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: $\chi_{378} (89, \cdot )$
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 \( 1 + (-5.59 - 3.23i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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