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Properties

Label 2-378-189.38-c1-0-16
Degree $2$
Conductor $378$
Sign $0.543 + 0.839i$
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  + (−0.984 − 0.173i)2-s + (1.73 − 0.0354i)3-s + (0.939 + 0.342i)4-s + (−0.334 − 1.89i)5-s + (−1.71 − 0.265i)6-s + (2.15 − 1.53i)7-s + (−0.866 − 0.5i)8-s + (2.99 − 0.122i)9-s + 1.92i·10-s + (−1.95 − 0.344i)11-s + (1.63 + 0.558i)12-s + (−0.332 − 0.396i)13-s + (−2.38 + 1.13i)14-s + (−0.646 − 3.27i)15-s + (0.766 + 0.642i)16-s − 2.62·17-s + ⋯
L(s)  = 1  + (−0.696 − 0.122i)2-s + (0.999 − 0.0204i)3-s + (0.469 + 0.171i)4-s + (−0.149 − 0.848i)5-s + (−0.698 − 0.108i)6-s + (0.814 − 0.580i)7-s + (−0.306 − 0.176i)8-s + (0.999 − 0.0409i)9-s + 0.609i·10-s + (−0.589 − 0.103i)11-s + (0.473 + 0.161i)12-s + (−0.0922 − 0.109i)13-s + (−0.638 + 0.304i)14-s + (−0.166 − 0.845i)15-s + (0.191 + 0.160i)16-s − 0.635·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.543 + 0.839i$
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: \((2,\ 378,\ (\ :1/2),\ 0.543 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23921 - 0.673509i\)
\(L(\frac12)\) \(\approx\) \(1.23921 - 0.673509i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 0.0354i)T \)
7 \( 1 + (-2.15 + 1.53i)T \)
good5 \( 1 + (0.334 + 1.89i)T + (-4.69 + 1.71i)T^{2} \)
11 \( 1 + (1.95 + 0.344i)T + (10.3 + 3.76i)T^{2} \)
13 \( 1 + (0.332 + 0.396i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + 2.62T + 17T^{2} \)
19 \( 1 + 2.64iT - 19T^{2} \)
23 \( 1 + (-0.904 - 1.07i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (-1.24 + 1.48i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (3.14 - 8.65i)T + (-23.7 - 19.9i)T^{2} \)
37 \( 1 + (2.52 - 4.37i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.09 + 3.43i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-10.5 + 3.82i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (1.15 - 0.420i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-2.98 - 1.72i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.01 - 2.53i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.959 - 2.63i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.405 + 2.30i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (7.92 - 4.57i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (13.9 - 8.02i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.00 - 5.70i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-11.7 - 9.85i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + (-0.655 - 1.79i)T + (-74.3 + 62.3i)T^{2} \)
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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

−10.94012845334941387208542231141, −10.27640955823009040385045894276, −9.039305799227803908664256920214, −8.616566067197840535805968931649, −7.71460592453064889463459120083, −6.96737006963454272074819218074, −5.12698038310690838600513309362, −4.12062750336312343422032035888, −2.61834428857252086476876186213, −1.19974477612277707854287403591, 1.98244340342988518532945557441, 2.92500303984272490489562630823, 4.43500364436598622685340116024, 5.93846308938246498245500067522, 7.23047213102314178045957013469, 7.80700291734157393053106456310, 8.719631945872244288908528261866, 9.501091032249854823381740331446, 10.57350646698799375812081364366, 11.16618268838008563169151583168

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