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Properties

Label 2-378-189.58-c1-0-13
Degree $2$
Conductor $378$
Sign $0.512 + 0.858i$
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  + (0.766 − 0.642i)2-s + (−1.07 + 1.35i)3-s + (0.173 − 0.984i)4-s + (−0.423 − 0.355i)5-s + (0.0485 + 1.73i)6-s + (0.945 − 2.47i)7-s + (−0.500 − 0.866i)8-s + (−0.685 − 2.92i)9-s − 0.553·10-s + (1.51 − 1.27i)11-s + (1.15 + 1.29i)12-s + (1.69 − 0.615i)13-s + (−0.864 − 2.50i)14-s + (0.938 − 0.192i)15-s + (−0.939 − 0.342i)16-s + 2.04·17-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (−0.621 + 0.783i)3-s + (0.0868 − 0.492i)4-s + (−0.189 − 0.159i)5-s + (0.0198 + 0.706i)6-s + (0.357 − 0.934i)7-s + (−0.176 − 0.306i)8-s + (−0.228 − 0.973i)9-s − 0.174·10-s + (0.458 − 0.384i)11-s + (0.332 + 0.373i)12-s + (0.469 − 0.170i)13-s + (−0.231 − 0.668i)14-s + (0.242 − 0.0497i)15-s + (−0.234 − 0.0855i)16-s + 0.497·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.512 + 0.858i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.512 + 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29332 - 0.733887i\)
\(L(\frac12)\) \(\approx\) \(1.29332 - 0.733887i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
3 \( 1 + (1.07 - 1.35i)T \)
7 \( 1 + (-0.945 + 2.47i)T \)
good5 \( 1 + (0.423 + 0.355i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (-1.51 + 1.27i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (-1.69 + 0.615i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 - 2.04T + 17T^{2} \)
19 \( 1 - 5.54T + 19T^{2} \)
23 \( 1 + (0.623 - 0.226i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (2.85 + 1.03i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (0.341 - 1.93i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (1.52 + 2.64i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.66 - 2.79i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.827 + 4.69i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-1.88 - 10.7i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-0.951 - 1.64i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.797 - 0.290i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.346 - 1.96i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-11.6 - 9.74i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (4.69 - 8.12i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.50 - 7.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-11.0 + 9.24i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (4.11 + 1.49i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 + (2.85 + 16.1i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.34247938723424615256141732073, −10.41392719906460753947217376055, −9.775445033639867020573427832324, −8.608954423172544188961688395632, −7.30311649259776191368735473302, −6.10561709049625640328219780980, −5.16282194804814012507321470793, −4.13652858892862995968994362279, −3.35949477579543192495712677208, −1.02689054437362160333048333817, 1.79062543872304434111975700259, 3.39797383337781589210222820641, 5.00997014868432781620005556226, 5.68788797679846817364553358084, 6.70910540724796950098971793949, 7.55540010465739525024255330154, 8.443585433181989632414685300065, 9.594150104621812209625174096011, 11.04256644234753759844398095765, 11.79681247189208558850805415227

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